Why did WWII aircraft carriers lack the angled decks now found in every modern carrier, and why were such carriers not designed and built during the war?

Angled decks were not used on WWII-era carriers because they had not yet been developed; they were not developed because they were not necessary.

All carriers designed before or during WWII used an 'axial' (straight) deck design (barring a few early experiments into multi-level flight decks, like that seen on the Courageous class carriers, or on Akagi and Kagi before their rebuilds). This deck was typically divided into two areas by a crash barrier. The barrier was a fence of steel wires, supported by collapsible stanchions, and could be raised or lowered as needed. The barrier was designed to stop aircraft that missed the arrester wires from travelling forwards into aircraft parked or operating forwards. On a ship with a barrier, aircraft could land on the after part of the hull, while operations were ongoing in the forwards part. If aircraft needed to land and take off simultaneously, they could be landed on the after part, while aircraft were being launched from the forwards part. If lots of aircraft needed to be landed at once, then after each aircraft landed, it could taxi forward of the barrier, and be parked in a deck park while awaiting maintenance or a movement to the hangar. If a large amount of aircraft needed to be launched at once, then the barrier could be lowered, and a full strike ranged on the full length of the deck. This system was efficient, and worked well for the aircraft commonly used during WWII.

As the war progressed, however, it became clear that aircraft sizes and speeds would increase to the point where the barrier would no longer effectively split the flight deck into two separate sections. Both the RN and USN wished to use large, twin-engined fighters to extend the striking range of their carriers, and to allow the effective use of new technologies like airborne radar. The RN would put this concept into production with the De Havilland Sea Hornet (and prototype Sea Mosquito), while the USN did it with Grumman's F7F Tigercat. These aircraft were fast enough, and heavy enough, to push the boundaries of what practical barriers could handle. It was clear that future aircraft would be larger, especially if it was necessary to carry nuclear weapons aboard them. The introduction of jets only exacerbated these trends. Jet engines required high landing speeds, were typically used on heavy aircraft, and had poor throttle response compared to piston engines (which meant that if a jet aircraft missed the arrester wires, it could not attempt to take off and go around again before hitting the barrier, as a piston engined aircraft might). These meant that the safety barrier was not able to effectively stop jet aircraft. It was clear that jet engined aircraft could not be safely operated from axial deck carriers without drastically reducing the carrier's ability to generate sorties or maintain a CAP.

In the late 1940s, Royal Navy engineers tried a number of options to allow the safe operation of jets without limiting the abilities of the carrier. A number of innovations were suggested, including vertically separated landing and take-off decks (as had been tried unsuccessfully pre-war on the Courageous-class). One innovation that the RN put a lot of time into developing was the 'flexible' deck. This was a rubber deck, on which an aircraft without an undercarriage could make a soft landing. The flexible deck would solve the problem of landing jets safely, and would also improve the performance of jets, by removing their undercarriage and hence reducing weight. As the flexible deck could only be used for landing aircraft, a separated launching deck was needed. Testing by the RN and USN in the late 1940s and early 1950s proved that the flexible deck could be safely used for landing aircraft, but that it could not meet the targets needed for landing aircraft sufficiently quickly. It required a few minutes to move an aircraft off the flexible deck, but the RN in particular required just 30 seconds between landings. As a result, the flexible deck had to be abandoned. However, some of the innovations behind it were recycled; the idea of a separated take-off and landing deck was developed into the angled flight deck by Captain Dennis Cambell (an RN officer) and Lewis Boddington, the Head of Naval Aircraft Department at Royal Aircraft Establishment, Farnborough. The concept was tested aboard the carrier Triumph in 1951, using a simulated 10^o offset deck painted on to her flight deck. It was also passed to the Americans, partly by the test pilot Eric Brown, who was in the US as an exchange pilot. The USN tested it aboard the carrier Midway in May 1952, again using a painted deck, before converting the axial-deck carrier Antietam to use an angled deck in 1952, with the first landings aboard her taking place in January 1953. In June 1953, the USN and RN held a joint exercise aboard Antietam in the English Channel, giving RN pilots their first experiences aboard a true angled-deck carrier. This experience finally convinced the Admiralty of the advantage of the angled deck, and all future conventional carriers used by the RN would have angled decks - the Illustrious and Queen Elizabeth class, built for STOVL aircraft, do not need an angled deck because aircraft landing on them land vertically, rather than using a rolling landing.