Was it far and away better than Allied/Soviet aircraft, or just marginally better? If it weren't for the production problems and fuel shortages (that is, had the Germans successfully deployed it in large numbers), might it have significantly altered the course of the war?
I suppose objectively it wasn't "better" at all, it had certain major advantages but it had major drawbacks as well.
Let's compare it to a top-rated contemporary allied fighter which went head to head with the ME-262.
The two are of a similar size and though the P-51 has a lower aspect ratio the ME-262 has more powerful engines and lighter frame which gave the two aircraft similar payload capabilities (with a slight edge to the ME-262).
The aircraft were armed in a similar manner. The ME-262 carried 4 30mm cannons compared to the P-51s 6 .50 cal machine guns. Both of these configurations were absolutely deadly, but the edge again goes to the ME-262 especially when facing heavy targets such as bombers. Both aircraft could carry rockets but they approached it differently, the P-51 carried 6-10 127mm rockets and the Me-262 carried 24 55mm rockets. Both setups were good for different things, but in the end this is close enough to call a tie. Lastly, there is bomb loads and we see that the two planes are close again. The ME-262 had the capability to take 2 500kg bombs into the air compared to the P-51s max bombload of 907kg. Another slight edge to the ME-262, but the two planes have remarkably similar weapons capabilities.
Now we get to where the aircraft started to differ: The ME-262 was insanely fast. A max speed of 900 km/h made it the fastest thing in the sky that wasn't a prototype, and the poor P-51 (despite being one of the fastest allied production fighters in regular service) could not keep up with its maximum speed of 700 kp/h. This is obviously a huge advantage to the Me-262.
The Me-262 also had a much higher rate of climb (20 m/s vs 16.3 m/s). The ability to climb away fast coupled with its overall speed made the 262 hard to catch if it wanted to get away.
In every other major aspect, the P-51 was superior. It had a slightly higher operating ceiling (12800 m vs 11450 m), a longer range (2,755 km vs 1,050 km) and, most tellingly, it was a much more maneuverable aircraft. P-51 pilots learned that although they had no chance of catching a 262 in a race they had a chance to do some serious damage if they flew ABOVE the bomber formations and dove in on the jets while they were making their attack runs. The 262s couldn't attack accurately at full speed, meaning the P-51s had a window to get on them before they could accelerate and climb away.
The Me-262 had some other problems. It required concrete runways to take off and land, which in war-torn Europe were few and far between. This allowed the allies to easily target 262 bases and make operations difficult. It also meant that the jets could not be easily deployed all over the theaters were Hitler to magically pull a few thousand more out of his arse.
The jet was also extremely vulnerable at low speeds. During takeoffs and landings the German pilots might as well have crawled out and painted targets on their aircraft. It was necessary for the Germans to build major flak defenses around all of the 262 bases to ward off allied fighters. Again, this made putting the jets in the right place at the right time practically impossible if the Germans magically had thousands of them.
In fact, the Me-262s were so vulnerable when they weren't in the air that of 1400 produced only 300 actually saw combat.
There were some other little issues with the wunder jet. It had this slight issue with breaking apart on its own now and then, killing the pilot. And those magnificently powerful 30mm cannons tended to jam an awful lot despite what you might here about glorious German engineering.
A last thing to consider is this: Every ME-262 pilot had to be re-trained to fly jets. Germany couldn't afford to pull experienced pilots out of the air to train them so late in the game.
So...was the 262 as good as people tend to think? Probably not. It was a fine aircraft and a finer psychological weapon, but there were ways to counter it and the allies were learning fairly quickly. Had the Germans had them in large numbers the 262s would have been a nightmare to deploy. The plane was definitely not used to its full advantage (being deployed as a fighter-bomber as opposed to a straight up fighter) and it could have taken an even deadlier toll on allied bomber formations, but it was all too little too late. By the time the 262 got swinging another thousand of them would have delayed the war by another couple months at best. The allied bombers were keeping parts and fuel from the existing ME-262s quite well, keeping another lot of them supplied would have been impossible.
Sources:
The Me 262 Stormbird: From the Pilots Who Flew, Fought, and Survived It by by Colin D. Heaton and Anne-Marie Lewis
Messerschmitt Me 262 in Defence of the Third Reich by Marek J. Murawski
P-51B/C Mustangs over the 3rd Reich by Tomasz Szlagor
Red Tails Black Wings: The Men of America's Black Air Force by John Holway
Jane's Fighting Aircraft of World War II
^^^Personal ^^^note: ^^^Some ^^^aircaft ^^^specifications ^^^do ^^^not ^^^line ^^^up ^^^exactly ^^^from ^^^source ^^^to ^^^source, ^^^mainly ^^^due ^^^to ^^^sources ^^^using ^^^different ^^^historical ^^^sources ^^^such ^^^as ^^^manufacturers ^^^specs ^^^vs ^^^military ^^^reports ^^^vs ^^^restored ^^^aircraft ^^^reports. ^^^When ^^^in ^^^doubt ^^^I ^^^have ^^^used ^^^my ^^^own ^^^judgement ^^^to ^^^choose ^^^a ^^^likely ^^^average ^^^for ^^^ease ^^^of ^^^reading.
The Me-262's effectiveness is actually rather difficult to determine. The first reason was that by the time it was deployed in 1944, the Germans were already losing ground. Couple it with aforementioned fuel shortages and the rather poor reliability of the Jumo 004, logistically the Me-262 would have made very little of an impact.
But before we delve into that, let's look at the technical aspects:
The Jumo 004: The engine itself wasn't a poor design at all actually. The fact that it could run on av-gas, Diesel or Coal derived fuel coupled with it's relative cheapness in comparison to the Jumo 213 and BMW 801 prop engines were quite the advantageous. The problem with the engines however was the lack of training for new pilots to adapt to the difference in responsiveness to the engines, the general unreliability, the large amount of maintenance and the inadequate materials to produce it would hamper proper development of the engine. Contrary to the popular belief, there actually was quite a lot of engines built, numbers estimate that over 6000 engines were built by the end of WWII. Quite an impressive number. So production issues weren't necessarily the problem as much as the engine itself.
What were the problems? Well for one, major overhauls were required between 30 to 50 hours of service. Which meant that 262s tended to be grounded after a couple of flights. Obviously this wasn't conducive to the effectiveness of the 262 regardless of fuel shortages. The engines like almost all other jet engines were prone to fodding (stones and debris being ingested therefore requiring turbine blades to be repaired). Furthermore the Jumo 004 required maintenance of combustion chambers every 20 or so hours. With this in mind, it's pretty apparent that the Jumo 004 required a rather high degree of maintenance. Most reports that detailed about the Jumo 004 mention that the average service time ranged from 10-22 hours so physically speaking it would have been impossible to actually keep a fleet of 262s up in the air.
Now your question poses the hypothetical of the Germans being able to field a large number of 262s. Oddly enough, the Germans did actually produce a sizeable amount, approximately 1400 262s. However again the problem comes with logistics. Approximately 200-300 were ever actually in operation despite the 1400 number. Couple it with the constant strafing and bombing operations done on proper and improvised German airfields in 1944-1945, the time consuming maintenance and the fact that only about 20% of the produced jets ever were operational and we see that 262 wouldn't have changed much because the air superiority could not be reached even with double the amount of jets in the air. One has to remember that the P-51s, Spitfires, Typhoons, P-47s, and Tempests combined would have outnumbered the Me-262 26:1 even if there were three times the operational and maintained 262s. And that's just an approximate number of fighters that were fielded by the USA and the British. This isn't including Soviet Bombers, Allied bombers, ground attackers, or any other planes. Numerically speaking the only way for the 262 to have made an enormous contribution to the war would be that the 262 entered the war in 1942, 100% of them were in service and not grounded, and that the pilots were 100% trained. Also the Jumo 004 would have to be at least twice as reliable to make an even decent effort to keep up numerically.
In other words, unless the entire war changed where the 262 was introduced 2 years prior with a much more reliable engine, not even three times the Me-262s could have made a difference.
Smith, J. Richard and Edward J. Creek. Jet Planes of the Third Reich. Boylston, Massachusetts: Monogram Aviation Publications, 1982.
Kay, Anthony L. (2007). Turbojet History and Development 1930-1960