There were several reasons why ships were built much more quickly during WWII than they were later in the 20th Century. A major one was that far more people were employed in the shipbuilding industry during the war than during the 1990s. At the peak of the wartime industry, some 89,300 people were employed by shipyards primarily building warships, with another 44,100 working to repair and convert ships. Another 102,400 people were employed building or repairing merchant vessels. To this must be added workers in similar trades; the Royal Navy's dockyards employed 36,700 people, and 88,900 worked in marine engineering. For comparison, in 1999, there were only ~27,500 people working in the entirety of the British shipbuilding and repair sector, a tenth of those working in wartime. To this must be added workers elsewhere in the British Empire. Canadian yards were the largest, employing over 82,000 men and women at peak employment in 1943. Australian yards were smaller, but still built some 113 ships for the Royal Australian Navy, as well as providing repair facilities for the USN and RN. The availability of this mass of manpower meant that much of the labour-intensive work of shipbuilding could be done at speed. Constructing a 'River'-class frigate took between 350,000 and 400,000 man-hours. An example yard that built them was Yarrows Ltd's Canadian yard at Esquimalt, which employed 2,470 people. Assuming Yarrows built ships one at a time, with the full workforce working on a single ship and using the upper bound of 400,000 man-hours, it would take the yard just under 162 hours (roughly a week) per ship. This is, of course, an unrealistic estimate (the fastest 'River' built in a Canadian yard took 5 months and 3 days to construct), but it is indicative of the speed of construction that could be achieved by leveraging manpower.

As well as having a larger workforce, Britain also had more yards available than it did in later periods. The last major class of warships built for the RN (before the 20-year-rule cutoff) were the Type 23 frigates. These were built at just two yards; Swan Hunter in Wallsend and Marconi Marine in Scotstoun. At the start of WWII, Britain had thirteen major warship builders, plus roughly 60 more yards focused on building merchant ships, some of which could be converted to building warships. Having more yards meant more slipways were available for shipbuilding, which meant that greater parallelism could be achieved. More ships could be built at the same time, increasing total throughput.

Finally, ships were considerably easier to build then than they are today. Modern ships are highly complex machines. They have vast electrical systems, which are needed by every one of their sensors and weapon systems. Those weapons can themselves have tens or hundreds of moving parts, due to increased usage of automation. Their engines are gas turbines, essentially jet engines repurposed for maritime use, much more difficult to produce than older steam turbines or reciprocating engines. A Type 23 has some 6 million individual elements to its construction. While modern construction techniques, such as the use of prefabricated blocks assembled inland, simplify the task, it is still a monumental achievement. The ships in service in WWII were considerably simpler to produce. Their electrical systems were relatively primitive, their armaments (especially on smaller ships) were simpler and easier to integrate to the hull, and they used simpler designs and systems. Many designs were also simplified to allow them to be constructed in yards that were more experienced in constructing merchant ships. The 'Flower' class corvettes are the most famous example of this. Their hull design was based on that of the whaler Southern Pride, built by the mercantile Smith's Dock on the Tees. They used simple reciprocating engines that could be built by any yard's engineering department. This simplicity meant that 'Flower's could be produced remarkably quickly, with the quickest taking just over four months to build. New techniques to simplify construction were introduced over the course of the war. Welding, which had been opposed by British yards and by unions, made significant inroads during the war. It was easier to teach to unskilled workers than more traditional methods like riveting, allowing a greater increase in the workforce. Prefabrication was introduced to the British industry during the war with the 'Loch' class frigates. These used smaller sections that could be easily produced by structural engineering firms that had not worked on ships before, mainly bridge builders. These sections were then transported to shipyards by rail, and assembled into the ship. Each ship had 1,360 units, which were produced by about a hundred subcontractors, with thirteen shipyards producing larger sections, assembling the ships and outfitting them.

The RN could also call upon the vast shipbuilding resources of the USA. As well as the methods outlined above, the US had access to a much larger workforce. American shipyards, many of which were built on new 'greenfield' sites, drew on this workforce. Using welding extensively meant that the workforce could be expanded massively. American designs tended to require more manhours than British ones, but with this greater workforce, the ships could be built incredibly quickly. American yards built hundreds of ships for the RN, including the vast majority of the RN's escort carriers.

There were many reasons for this, and you're in luck as I'm in the middle of a study on this topic (specifically geared towards destroyer escorts, but they touch on almost every aspect of production).

First, Britain did not build 65 aircraft carriers during WWII (which I will define as commissioned between 1 September 1939 and 1 September 1945). The British completed six fleet carriers, seven light carriers, one maintenance carriers^(1), and six escort carriers (all converted from merchant ships, some before and some after completion), or merely 20 carriers. The United States converted another 38 escort carriers from merchant ships for the British, 34 of these known as the Bogue/Attacker/Ruler class. I don't knot where your 65th comes from, perhaps USS Charger as she was HMS Charger for two days.

This underscores two points critical for your British question. First, many of the 885 "large ships" (what was your source and how do they define "large ship?") were not built in Britain. The US built them for the Royal Navy, as we did many other ships. For example, the US completed 78 Captain class frigates (in the US called Evarts/GMT and Buckley/TE class destroyer escorts) for the Royal Navy, and we gave them dozens of old destroyers, submarines, Coast Guard Cutters, via lend lease and hundreds of brand-new landing craft of all types (including several that must be included in your large ship number). In general, Britain built their own major combatants, but America built many of their smaller combatants.

Second, not all of these ships are created equal. Since you called out carriers, let's stay there. As I mentioned, there are three very broad groups of aircraft carriers. Fleet Carriers are the stereotypical carrier, large, fast, intended to engage enemy fleets and (in this case) all the new-built carriers had armored flight decks and hangar sides. Light Carriers (or as the British called them "Light Fleet Carriers") were smaller, as fast as a fleet carrier but with fewer aircraft and munitions, designed to supplement fleet carriers and attack secondary/less defended targets, and in this case they lacked armored flight decks and hangar sides. Escort Carriers were smaller still: many (including all the British ships) were converted from merchant ships and were slow, had few aircraft, had difficulty operating more modern aircraft^(2), and were design primarily to escort convoys, support troops ashore, hunt submarines, and ferry large numbers of aircraft from Point A to Point B. The only warship these ships were ever expected to see were submarines, while fleet carriers were protected against gunfire from enemy ships. However, escort carriers can be built far more quickly than fleet carriers, and this trickles down to other ships: rapid construction was the entire point of the destroyer escort program.

This in turn means that escort carriers can be built at shipyards that have no experience building warships. Commercial ships have far lower standards than proper warships, for the simple reason they are not expected to be shot at. Warships are not only designed to be shot at, but survive significant damage without sinking. This in turn means only some shipyards are able to built warships, especially top-tier combatants, so are limited to second string warships or merchant ships. For example, of the 17 US Shipyards that built destroyer escorts, only five built Fletcher class destroyers and three Sumner and Gearing class destroyers. Even here there is some evidence that some shipyard quality varied, with some shipyards building subpar ships, but I need to spend more time here before discussing this in detail for WWII (though it is clear for WWI ships, especially US Flush Deck Destroyers).

Now we turn to the other side of your question: how could we the Allies (mainly the US) build so much? In short, careful planning, massive facilities, and sacrificing capability.

The true secret was the War Production Board, which oversaw all aspects of US production and guided us through several bottlenecks. Last night I found this report, and while I have not had time to dig into in yet it discusses several of the difficulties in detail, especially working around shortages of many components. For example, US diesel engine and turbine production was a serious bottleneck, so Liberty Ships used large reciprocating engines, which were available in large numbers and did not impact other warships. This report also discusses some such issues, and if you really want to get into the weeds I recommend reading them. For a single example, take this memo chain on the Midway class Large Carriers, which states in part:

As to other types of ships which we could more profitably build using the steel which would go into the big carriers, the present building programs are, in my opinion, very well balanced as they stand. I personally have confidence in the 6000-ton, 5-inch cruisers of the ATLANTA Class. We have seven of them on order but the scheduled deliveries stretch out until the summer of 1945, largely because of the difficulty in meeting the delivery requirements for main machinery.

Actually today, and the situation will apparently continue, hull steel, particularly plates, is very much less critical than is manufacture of main and auxiliary machinery for ships, especially large reduction gears.

The special treatment (ballistic) steel is still tight but even that will be far less troublesome than it has been during the past year when enormous demands came in for alterations to ships in the fleet as well as for the building program.

Actually, the two big carriers will help to balance the materials situation because they will require only eight sets of main machinery (four shafts per ship) against a large weight of hull steel and armor.

My answer to the second question is therefore that the programs for other types appear to be adequate and, further, that a material increase in numbers of ships of other combatant types would throw us out of balance as to machinery deliveries.

One of the secrets was the rapid expansion of US shipyards just before WWII. The US created several Emergency Shipyards and funded the expansion of several others (Federal's Newark yard is not on this list). More shipyards means more building slips, some of these could built two dozen small ships at a time, freeing the larger shipyards for more important orders. For example, in early 1942 Landing Craft of all types became very high priority, and based on an analysis of the construction dates and building slip images even replaced carriers, cruisers, and destroyers at Bethlehem Quincy, Newport News, New York Shipbuilding, the Charleston Navy Yard, and the Boston Navy Yard, and I have not yet done this detailed analysis for the Philadelphia and Norfolk Navy Yards (where the incomplete battleship Kentucky was launched weeks after construction started to clear her building slip for LSTs). However, LST production soon shifted to the Ohio River, and apart from a few orders later on (most when other orders had been canceled) they built almost all that followed. Many of these became British ships, and I'm sure they are in that 885 large ship number.

Finally, where necessary, we sacrificed capability, especially early in the war. This is where destroyer escorts really touch on everything: the main difference between the six US classes were the bridge shape, hull length, and propulsion plant^(3). Two (Evarts/GMT, short hull, and Cannon/DET, long hull) classes used a diesel electric plant: four GM diesel engines driving electric generators that powered electric motors that drove the ship. Two more (Buckley/TE, 3"/high bridge, and Rudderow, low bridge/5") used turbo-electric drive, the same idea but with turbines and higher speed (rated for 24 vs. 19/21 knots depending on the official document). One (Edsall/FMR) used Fairbanks Morse diesels driving the propellers through a set of single reduction gears: more difficult to manufacture but necessary as electric motor production was critical. The last (John C. Butler/WGT) used turbines through a set of double reduction gears. All of these were necessary due to the production bottlenecks, and as those bottlenecks eased many ships planned to be 19/21 knot diesel ships or 3" turbo-electric ships to be upgraded to 5" guns.

This is repeated throughout US production. Submarines used a wide variety of diesel engines and reduction gears from several manufacturers due to supply difficulties, and while I have not spend as much time going down the Ship's Data for other types of ship even a cursory glance shows this held up and down the scale (especially down). I am only just scratching the surface of this very interesting corner of history, and there are many questions I still cannot answer.

While I can't specifically speak to details of British yard manufacturing, as far as the cargo ships, Churchill had this to say in his Closing the Ring at the beginning of 1942:

"The foundation of all our hopes and schemes was the immense shipbuilding program of the United States."

For some context as to what he meant, the "Happy Time" of the U Boats in mid to late 1940 alone had taken out ~1.5 million tons of British shipping, the successful execution of which was essentially the unimplemented German plan during World War I to cut Britain off at her knees by destroying her supply lines. It nearly worked; add in the 500k additional tons lost to sources other than U-Boats, and not only did this vastly exceed the replacement capacity of British yards - which even in 1939 had realized they didn't have the capacity to meet demand and were leisurely chugging away at around 750k tons per year for cargo production in the late 30s, slowly increasing to around 1 million by war's outbreak - but until 1943 there were genuine shortfalls of war materiel making it to British shores. (A great, quintessentially British line describing the disastrous conclusions of planners in 1937 and 1938: "British shipping is adequate for the first year of the war.")

By 1940 and 1941, this meant FDR had ordered the Navy to forceably bend all sorts of non-belligerent rules in terms of convoys and areas of responsibility (pushing the boundaries of 'neutrality' patrols all the way to Iceland was certainly interesting, especially after orders came down to shoot first if threatened) to try to get more cargo through, but with yards full for warship construction, the British realized the only hope of getting more capacity was to come hat in hand to the Canadians and the United States and try to buy cargo ships.

Problem was, even those yards were already largely full, and the initial emergency shipbuilding programs on both sides of the Atlantic didn't do all that much on a net basis for cargo ships despite adding roughly 1.5 million tons more per year production on the British side and a few hundred thousand tons on the American and Canadian side by late 1941. Once the Enigma machines got a fourth coding wheel in early 1942 and became unbreakable again for 10 months, it led to the second Happy Time when things got drastically worse. From Behrens via the Johnman and Murphy article:

British shipping net losses, 1942 tonnage

Quarter	Built	Sunk	Net
Q1	546	757	-211
Q2	607	892	-285
Q3	822	980	-158
Q4	626	1334	-708
Total	2601	3963	-1362

What saved the British? The entry of the Americans into the war, which allowed the Kaisers and Bechtels of the world to construct new yards (Bechtel had bulldozers clearing a newly purchased area next to San Francisco Bay literally 10 days after the Maritime Production Commission got the request to dramatically increase production in March 1942) and gave them the freedom and funding to pretty much hire anyone with a pulse who'd walked into their yards and immediately put them to work building ships for the merchant marine. This was the other part of the equation of that year and was a huge reason towards why the British weren't knocked out of the war, because even a few months after that initial request, production had begun to spike:

US controlled shipping net gains, 1942 tonnage

Quarter	Built	Sunk	Net
Q1	300	296	+4
Q2	909	587	+322
Q3	1387	490	+897
Q4	1727	287	+1440
Total	4323	1660	+2663

By 1943, with the Enigma code once again cracked and losses substantially declining by that spring with the end of the second Happy Time, the British added around 500k net tons to their merchant fleet despite production actually declining a bit. The Americans, though, had begun playing on an entirely different level. Despite having almost a million tons of shipping sunk in 1943, the new yards added a net 9.2 million tons of shipping - or far more than the Germans had sunk during the entire war up to that date.

I won't go into much detail about what made the Liberty yards so productive; there's an older thread with an answer by /u/ggorgg that discusses some of the work conditions, but essentially the contractors were allowed to do pretty much whatever they wanted labor wise in them, where women and African Americans found good employment for the first time in their lives. This has led to a lot of academic work on the labor and social effects of the yards, and even the healthcare aspects - Kaiser was one of the first employers to implement what was effectively an HMO to keep people healthy and at work, which still exists today as the largest healthcare provider in California, still named Kaiser Permanente after two of his facilities - but there hasn't been a ton written academically on the actual production methods since the seminal survey of Ships for Victory in the 1950s. I've only so far skimmed the one relatively recent (2003) PhD thesis that seems directly related, Tavassa's Launching a Thousand Ships, but it looks like it might answer any remaining questions on the cargo ship side along with a couple of the other sources I've listed.

Sources:

Johnman and Murphy, The British Merchant Shipping Mission in the United States and British Merchant Shipbuilding in the Second World War, Elphick, Liberty: The Ships that Won the War, Herman, Freedom's Forge: How American Business Produced Victory in World War II Paperback, CBA Behrens, Merchant Shipping and the Demands of War, Foster, Henry J. Kaiser: Builder in the Modern American West, Lane, Ships for Victory