Your note on the naval side is full of inaccuracies. The aircraft carrier predated WW2, having been developed by the Royal Navy in 1917 with HMS Furious, and improved a year later with HMS Argus. The Japanese developed their first carriers through observing British practice. It's also worth noting that the aircraft carrier was nowhere near perfected by this time. Allied navies made more significant steps towards the modern carrier. The angled flight deck, optical landing systems and the steam catapult, all required for jet operations, were developed by RN and USN officers towards the end of WW2. In comparison, Japanese carriers did not introduce anything of significance to carrier warfare.

Submarines are vastly important to modern naval warfare, but this isn't entirely because of guided missiles. The Germans had three main developments in WW2 that were highly relevant to submariners - the acoustic torpedo, the HTP air-independent propulsion system, and the streamlined, tear-drop shaped hull. Of these three, the acoustic torpedo was also independently and simultaneously invented in the US, with both the Mk 24 mine and the G7e/T4 entering service in March 1943. The HTP AIP system was developed in Germany, but post-war trials proved it far too dangerous to be useful - the British submarine Meteorite using this engine was officially deemed "75% safe". The Germans introduced streamlined hulls on the Type XXI submarines, but these had been first implemented on the British R-class submarines of 1918, which prefigured the modern hunter-killer submarine. The R-class was a design dead-end, but this was mainly due to poor battery technology and surface performance. The Type XXI submarines were highly influential on post-war submarine design. However, the more significant technologies for submarine warfare included the nuclear reactor, the snorkel (a Dutch invention), and the computer - these allowed for submarines to spend far more time submerged, and make far more accurate torpedo shots. The allies also made significant advances in technology for anti-submarine warfare - the sonobuoy, MAD systems, airborne searchlights, and acoustic countermeasures were all Allied inventions of WW2.

You've also completely neglected Allied innovations in radar and aerial refueling. The British invention of the cavity magnetron allowed for massive reduction in the size of radar systems. By the end of the war, they had radar systems that could fit into artillery shells - the VT proximity fuse. They produced early ground-mapping radar, the H2S, the first aerial warning and control aircraft in the TBM-3W and PB1-W, the first air-interception and surface search radars and the first IFF system. Aerial refueling was pioneered by the USAAF in the 1920s, and was heavily developed by Sir Alan Cobham's Flight Refueling Ltd. By 1939, British Shorts Empire flying boats made 15 crossings of the Atlantic, relying on aerial refueling. Plans were in place for British bombers of Tiger Force to bomb Japan using in-flight refueling, but these were brought to an end by the end of the war. These technologies have had a massive impact on modern aerial warfare, just as much as the invention of the guided missile or jet engine.

Edit:

Though to answer the question in the title of this post, one of the big things that slowed Allied R&D on things like jet aircraft was a concentration on production of existing, working products over development of new ones. Frank Whittle's jet engine design could have been brought into production much earlier had it been worked on by Rolls Royce throughout. But Rolls Royce were heavily focused on the Merlin engine, and so early R&D work on a British jet engine was done by the comparatively poor Rover. Similarly, the work of putting the engine into an aircraft was done by the second-line Gloster, rather than the more experienced Hawker and Supermarine, as Gloster wasn't producing anything critical.

First use of helicopters in military operations: 1, 2

Two problems. First, the differing standards for each item. Here, first operational use counts. Other times, first prototype is all that matters. Second, which goes to the heart of the post, is the issue of parallel and independent development. To put it simply, it's not enough to be first to claim you're responsible for modern influence. For example, the V-2 undeniably had major influence on ballistic missile development. But what influence did German helicopter use have on the US as you claim when they had already invested in Sikorsky's work before that German operational use, and indeed American helicopter use would exceed Germany's?

First adoptee of jet technology: Me 262

Again, the Germans were first, but by a matter of months. The British already had a jet fighter in development and approaching service. If you want to argue influence you have to show it, like for the Me 262, the influence of swept wings.

Modern LMG concept*: MG34

The MG 34 was a general purpose machine gun. It was the first of that, but the modern use of GMPGs doesn't actually follow the German use of them as a squad based weapon. Instead Western armies follow a fireteam based SAW model (and not to break the 20 year rule, but your dismissal of mere automatic rifles is ironic considering the proposed resurgence of them today).

First to use remote-control unmanned vehicles in a military capacity: Goliath

Right country, wrong war. The Imperial German Navy was using remote control boats against the Royal Navy in WWI.

Tactical guided missiles: Antiship , Antitank, Antiair

Again, parallel and independent development. The US was developing guided weapons before it ever encountered any German ones. You have to look at it at a lower level than simply PGMs. Some, like their work on wire guidance, was very influential. On the other hand I'm not aware of any German prototype infrared or radar guided weapons, which so dominate the modern world, while the US prototyped the former and actually deployed the latter.

Modern armor doctrine: Again, this is very debatable; Soviet deep battle is similar but was not adopted until after the use of the 'blitzkreig' armor concentration doctrines

Actually Deep Battle was adopted and then abandoned before WWII. You might claim then it doesn't count, but again I put to your examples of prototypes. Indeed blitzkrieg (not to get into the whole debate of what exactly it is) was influenced by Western ideas, which were mostly never adopted at all in the pre-war time period, but again, what counts as first?

Stealth Aircraft, also first flying wing: Ho229

Leaving aside the merits of the claim of the Ho229's "stealth" if we accept it, there's no evidence that Lockheed/USAF/CIA even knew about it in the 1950s when they first seriously considered RCS reduction. As for flying wings, Jack Northrop started building them in the 1920s, both independently and as far as I know before the Horten brothers.

Heavy Airlift airframe (IIUC): Me323

What exactly qualifies that as "heavy airlift"? It's certainly heavier than a C-54, though its margin in term of usable payload is rather meager. And again, what was its actual influence, compared to say the C-97 and C-74?

Cornershot concept: Krummlauf

The Cornershot (whose ratio of relevance to modern warfare compared to Discovery Channel appearances is rather low) not actually a bent barrel. And if you want the general idea of firing a weapon from cover, then you could just as soon trace the Krummlauf as a development of periscope trench weapons of WWI.

SAMs: Rheintocher

US Army and Navy SAM development began before anyone saw this (in fact the USN set a requirement for a SAM as the natural counter to your already cited Fritz X) and what they came up with (Ajax and the 3Ts) bare no resemblance to it, so again, where's the influence?

Very long range bomber (IIUC): Ju89 'Ural'

Again, what is the line between a long range bomber and a "very long" range bomber? Especially considering heavy bombers existed before the Ju 89 and again where is the influence?

Their investment in long-range rockets was completely counterproductive. One way to think about the V programs is that they reflected a feeling that the propaganda was better than the results, that they were pushing the technological edge into places that it was not actually productive, and that the best you can say about it is that when paired with other, better technology (like atomic bombs) they could be relevant.

Now, we can ask, why'd they sink so much money into that particular hole? The answers here are someone illuminating. Part of it was a belief that fantastical weapons would produce a disproportionate propaganda effect. This turned out to be false. But the centralization of the German state meant that the views of a few at the top could set priorities all the way down, even if a good military review board would have looked at it and thought it not very promising (at least not for the investment involved). There were fantastical rocket visionaries in many countries, but Germany was the place where those visionaries managed to inspire people who wielded immense power.

Another salient point is that this technology could be worked on in the interwar period without violating the Treaty of Versailles. This subtly pushed the Germans into several areas that they might not have otherwise immediately worked on — it meant that they could start investing in things early on without fear of sanction. (Later, obviously, they started to explicitly re-militarize.) They also put a lot of hope into "hail mary" sorts of technologies that would make winning the war easier on them.

Were they really light-years ahead in the technology? No. They were on the edge of what was possible at the time, if you threw money at it. The German state was good at throwing money at defense projects. The United States was, until about 1941, not that good at it — even after the experience of WWI there were many who didn't value the way that new contributions of science and technology could affect the tide of warfare. The bureaucratic mechanisms for hooking scientists up with industry and the military were being developed somewhat on the fly. Even the Office of Scientific Research and Development (created in 1941, given broad powers to centralize scientific research and develop it into actual weapons) was relatively weak in many ways. (The US got better at this during the Cold War, if by "better" one means, "threw a lot more money at the problem.") The Germans were pretty good at this already in the 1930s, in part because of the aforementioned beliefs of those at the top that this could be decisive.

In many ways, the Germans bet their money on the wrong horses, technologically speaking. They weren't entirely wrong about some of the choices paying out in the long-term, but they were wrong about them making big differences during the Second World War. Some of the choices made by the UK and US were in retrospect much better — notably radar, the proximity fuse, decryption computing, and, of course, the atomic bomb.

As for the Americans and Soviets picking up where the Germans left off — indeed. Spoils of war, plus some of these technologies, given another decade to mature, were really important technologies. But not all of them could be made that important in WWII.

Should we be in awe of the Germans? Only if we are in awe of people who get sucked in by hype. There is nothing valorous about throwing a lot of money at problems that can't be solved by the time you need them solved, especially at the expense of throwing it at problems that can actually contribute to near-term success. This is a big generalization but for some of these technologies in question I think it is justified.

OP, you seem to be laboring under a few misapprehensions.

First of all, you seem to have a poor familiarity with advances made by powers which were not the Nazis; for instance, the USN pioneered a highly networked computerized approach to anti-aircraft warfare, and of course, computers are still in use with navies today. Another example would be lighter support weapons-- the Germans pioneered LMGs, but that's not the only support weapon used, and the USMC is transitioning back to magazine-fed weapons more reminiscent of the BAR or Bren.

Secondly, you seem to give "advancements" to Germany, even when both sides made them. For instance, the US had some drones as early as 1918, and had operationally-useful glide-bombs in WW2, as well. As both sides used them, I don't see why you credit the invention of the glide bomb to Nazi Germany; after all, they did not come up with the idea, or with prototypes, and the US started working on Azon and Bat long before the Germans demonstrated theirs. If the Allies started before they knew about the German weapon, the Germans hardly could have inspired them, could they?

Finally, if something was tried by Germany, dropped for a long time, and then was restarted, you can't really claim that the restart was due to the German example. For instance, flying wings were prototyped more-or-less simultaneously by the US (YB-35 at an advanced level in 1946, but Northrop designs flew pre-war) and Germany. However, they were then dropped, because they were too unstable. Only with the invention of radar stealth were flying wings built, because computers gave more stability, and radar stealth motivated the designs. Just because the Horton might have been stealthy doesn't mean that the Germans invented radar stealth; the aircraft wasn't designed with stealth in mind, and the Germans likely would never have discovered its mild low-observable characteristics.

The same goes for helicopter assaults--they weren't adopted based on the experience with a few Nazi commando raids, they were used as equipment permitted. The USMC conducted an assault in Korea, the UK conducted a larger assault 5 years later in Egypt, and the French conducted many assaults in counter-insurgency operations, but only when helicopters had gotten much better did air assault become commonplace. Until then, helicopters were used for liaison, resupply, medivac, and ASW, just as they were used by the allies at the close of WW2.

I will say that there is a trend to exaggerate the effectiveness of the other side's weapons. For instance, we always hear about how terrible the M4 sherman was, and how great the tigers and panthers were--and yet, German super-tanks were notoriously unreliable and thirsty, and only a small number were built. The M4 proved superior because it was reliable, and perfectly capable of engaging enemy tanks in the numbers available. Everyone wishes for superior equipment, but equipment with superior characteristics on paper is often more temperamental and prohibitively difficult to produce.

Be wary of seeing an idea on paper and thinking "aha, the Germans invented this idea!" Just because the Germans had an idea for a trans-atlantic nuclear bomber doesn't make it practical; even the operational B-36 wasn't really practical. What you don't see in the design proposal is all of the flaws--the unreliability, the basing limitations, the vulnerability to interception. After all, the allies had a whole pile of wild-eyed ideas which seemed ahead of their time.

You're point about the zielgerat ignores that the United states also had a infrared scope in service in 1945 with the M3 Carbine; some were issued to Marine and Army units on Okinawa.

Moreover, Deep Battle developed alongside so-called 'Blitzkrieg' in the 20's and 30's, and was pretty much the standard school of thought, until the likes of Isserson, Triandafillov and Tukhachevsky were purged in the thirties; by 1943, much of it was being re-learned, and was demonstrated in 1944 (i.e. operation Bagration).

The seemingly advanced development of German science and technology under the auspices of the Third Reich is complicated matter and reflected both the ideological imperatives of the regime and a long-standing research infrastructure within Germany that predated the rise of Hitler. The state's investment in scientific research that was applicable to military technology often dovetailed with the research infrastructures that had evolved inside Germany. When it was not hounding out political and racial undesirables in industry and higher education/technical institutes, the Third Reich forged a practical relationship between the state, industry (often producing for the military), and the various research organizations.

The Third Reich saw German research and scientific development in the applied sciences as a sign of German racial superiority and genius. The state often lavished funding and research grants upon existing institutions. This meant that scientific establishments within Germany, which had already established a world-leading reputation in the early twentieth century, grew ever larger under the Third Reich. Aviation in particular received a great deal of support from the state. It founded the Nationalsozialistisches Fliegerkorps (National Socialists Fliers Corps) which co-opted many existing aviation clubs into a central umbrella organization that encouraged air-mindedness amongst Germany's youth. Aviation concerns like Heinkel and Messerschmitt found themselves working in partnership with the newly established Reichsluftfahrtministerium (RLM- Air Ministry) which gave them access to funding and other perks of access to government. It should be noted that not all aviation industrialists liked the new order, Hugo Junkers, a prominent socialist and pacifist was put under house arrest in 1933 and had to sign over ownership of his company and his patents to the state. But many other aviation firms found partnership with RLM to be quite lucrative for the development of new aircraft. For example, German aviation researchers had some 63 wind tunnel complexes available to them within the Reich to develop and test high-speed design. In contrast, the US had only 3 major wind-tunnel complexes and were not as powerful as the German ones. Of course, German science was not omnipotent and there were considerable oversights and underfunding of areas the Third Reich did not deem important. Physics, for example, was at least in the 1930s, not much of an applied science and enjoyed less prestige in the Weimar Republic and Imperial Germany. This was why many German Jewish researchers gravitated towards physics. The academic establishment, which could be quite antisemitic, blocked off advancement into more prestigious scientific fields. This led to a brain drain in these departments, which given the importance these emigres had for the Manhattan Project has led to a perception that the Third Reich underfunded the sciences and was left with second-stringers. Yet as a whole, much of the applied sciences fared quite well under the Third Reich despite the state's political repression.

However, research in advanced technology does not necessarily translate into a useful end product. The proliferation of prototypes and special designs was as a whole counterproductive to the German war effort. The state also compounded this problem considerably. The polycratic nature of the Third Reich in which access to officials in the NSDAP meant that sometimes funding went into R&D projects that were technological dead ends. The Horten Ho-229 is an excellent example of this phenomenon. Goering believed that the Horten brothers were aeronautical geniuses and lavished funding on their flying wing designs. Although interesting from an experimental standpoint, the Horton Ho-229 was just not a very practical aircraft. It had the issues common with flying wing designs that would not be resolved until flight instrumentation would catch up in the 1980s and some other issues unique to the design itself like placing the pilot between two loud jet engines. Its purported stealth features were relatively minimal and the Horten brothers were obsessed with the flying-wing as an aerodynamic concept, not as a means to evade radar and its wooden construction was a weight and materials saving feature. As the Horten case exemplifies, access to this expanded scientific establishment was often predicated upon connections to the state and membership in the NSDAP helped in these matters. By mid-war, Goering was quite frustrated with his conventional designers, especially after the costly failures of the Me-210 and the persistent problems of the He-177, and the Horten brothers were an alternative to design bureaus that had failed in Goering's estimation.

Although the German armed forces as a whole was committed to high quality weaponry, the design approach of the Wehrmacht was actually rather conservative when it came to implementing new designs. This was partly the legacy of the Weimar Republic's Reichswehr which under the auspices of Hans von Seeckt outlined a relatively detailed plan of expansion once the Versailles restrictions had been ended. These plans incorporated some of the lessons of the First World War and did inculcate a progressive view on some technology. The German commitment to man-portable heavy machine guns was an example of this type of thinking. However, the Waffenamt which was the testing and procurement bureau, adapted a cautious approach to new weaponry. There was very little pressure to move beyond bolt-action rifles as the main weapon for the infantry. Perhaps most shocking, the Waffenamt remained decidedly behind the curve of the explosive growth in armor and the need for heavier antitank weapons. The Reichswehr's reforms had emphasized portability, so the Germans entered the war with a main antitank gun, the 3.7 cm Pak 36, which was clearly inadequate for its role despite receiving intelligence reports about French and British armor developments.

The experience of war forced the Germans to adapt stopgaps and push for the development of technology before it had matured. Some of these efforts proved to be quite effective. The Panzerfaust and Panzerschreck came out of reports from North Africa of the bazooka and both became cheap single-man portable antitank weapons with no Allied counterpart. However, these forays into the future should not obscure the fact that the bulk of German equipment was increasingly outdated as the war progressed. The German research apparatus tried to arrest this trend, but with mixed results. The Bf-109 is a case in point. Although a world-leader in 1939/40, albeit with some nasty handling characteristics, the design was reaching the end of its limits in 1941/2. To meet the needs of aerial warfare, Messerschmitt installed a larger powerplant (the DB 605) and tweaked the airframe, but this in turn created its own set of problems. The oil-cooling of the G/K was only barely adequate and the G/Ks were simultaneously overarmed and underarmed; the center line cannon's slow muzzle velocity and tendency to jam limited its effectiveness, and the two MG 131s were reliable, but unable to put a lot of throw weight onto a target. Adding more tweaks to the airframe to resolve these issues added more weight, which stressed the engines and made the handling characteristics of the plane worse. The plane had reached the near end of its development cycle and there was no magic bullet to arrest this.

Sources

Corum, James S. The Roots of Blitzkrieg: Hans Von Seeckt and German Military Reform. Lawrence, Kan: University Press of Kansas, 1992.

_.The Luftwaffe: Creating the Operational Air War, 1918-1940. Lawrence: University Press of Kansas, 1997.

Deichmann, Ute. Biologists under Hitler. Cambridge, Mass: Harvard University Press, 1996.

Habeck, Mary R. Storm of Steel: The Development of Armor Doctrine in Germany and the Soviet Union, 1919-1939. Ithaca: Cornell University Press, 2003.

Schmoll, Peter. Nest of Eagles: Messerschmitt Production and Flight-Testing at Regensburg 1936-1945. Hersham: Ian Allan, 2010.

Tooze, J. Adam. The Wages of Destruction: The Making and Breaking of the Nazi Economy. New York: Viking, 2007.

Uziel, Daniel. Arming the Luftwaffe: The German Aviation Industry in World War II. Jefferson, N.C.: McFarland & Company, 2012.

All sides invented lots of things, I don't see how cherry-picking some of the inventions of one side is supposed to show anything... And some of those things haven't been used by the allies for very good reasons.

• Jet engines: every major power was working on them, Germany simply deployed them in combat slightly earlier out of desperation.

• Stealth aircraft: Even if the Horten flying wings had actually been that stealthy in practice, it would still have been completely coincidental.

• Balistic missiles: again, everyone was working on rockets, the allies simply didn't waste their time and ressources on using them for strategic bombardments during the war, since it had very little effect. (Though artillery rockets were used by all sides.)

• Rocket powered aircraft: again, the allies didn't need them.

• Ultra long range bombers: the Germans build two prototypes, the Americans designed, produced and deployed a far superiour airframe in the B-36.

• Cornershot: pointless.

• Nerve agents: pointless.

• SAM: pointless, since there where too many bombers flying at the time.

I think you're cherry picking. Both sides invested in research, but the Allies happened to gain the upper hand in two very important technologies, radar and code breaking.

In radar the Allies had the advantage of high frequency, all of the main microwave power amplifiers currently in use were invented in the USA and UK shortly before or during WWII: the klystron in the US in 1937, the traveling wave tube (TWT) in the UK in 1942 (by an Austrian Jewish refugee), and the magnetron in the UK in 1940.

In computers, although the German did some pioneering work in hardware, they lagged behind in software development. According to some estimates, work done on code breaking at the Bletchley Park center may have shortened the war by two years. Alan Turing, the most famous scientist working there, is widely regarded as one of the founders of computer science.

Just a quick question: "First use of helicopters in military operations" - what do you mean by military operations in this case? Both of the examples you mention were used in a role of spotters/reconnaissance or be planned in the possible role of battlefield transportation yet not used directly in military operations.

A lot of people came a dumped on you a bit here, poking holes in your ideas and observations, but don't feel bad--you asked a great question.

With the exception of their Blitzkrieg Doctrine, most of the weapons cited were barely in testing when the war ended or were failures in operations. The V-2, for example, was a horribly expensive war to get 2000 lbs of explosives somewhere within 20 miles of your target. The ME-262 engines lasted between 25 and 50 hours before requiring a complete rebuild. The Nazi research system was horribly managed, and many projects were developed on political whims.

First to use remote-control unmanned vehicles in a military capacity: Goliath

Quite inaccurate. Soviets experimented with unmanned tanks since the late 20's, first Renault FT, then Soviet made unmanned T-18, then T-27, T-26 and BT-7. Soviet army had battalions (at least two) of unmanned tanks prior to WWII, for example 152th separate tank battalion (equipped with TT-26 tanks - unmanned flamethrower tanks, and TU-26 command tanks, which were controlling unmanned tanks at 1000-2000 meter distance).

The Bren gun was certainly not just an automatic rifle, as the BAR was (and a dreadful one at that!).

The Bren was fully automatic, had a quick change barrel and was quick and efficient to load during rapid fire. It could also be fitted to a tripod, an anti aircraft mount and was mounted on light armored vehicles like the Universal Carrier.

The Madsen, ZB26 and Lewis gun are all examples of light machine guns in service. There are many more.

Lastly, the MG34 was a GPMG. It could be used in a LMG configuration with a bipod or mounted to a tripod in the HMG role.