The first ships with a well deck were the USN's Ashland class LSDs, followed closely by the RN's 'Way' class. These were built in America to a design by the RN's Rowland Baker. There were a number of other landing-craft carriers that came into service earlier in the war. The various LSIs and APAs carried, and launched infantry landing craft from davits, while the RN developed a number of extemporised solutions for launching tank landing craft. A number of train ferries were converted to Landing Ship Stern Chutes, launching LCMs down a chute at the stern, while the Landing Ship Gantries were tankers converted to carry landing craft and launch them via a large crane. These made it much easier to transport and land tanks for the early waves of an amphibious assault. LSTs were too vulnerable for an assault, while LCTs/LCMs were not seaworthy enough to travel long distances over open sea, especially when loaded. Craning tanks into a landing craft from a larger transport was slow and dangerous. Being able to transport the LCTs aboard a larger ship and load them at the target meant that the first wave of an amphibious assault could benefit from armoured support.
An early LPD-style vessel is unlikely; it requires you to compromise both halves of the ship for little gain. Landing-craft carriers require a lot of space for the craft, their crews, and the troops to be transported aboard them. Aircraft carriers, even escort ones, require a lot of space for aircraft, fuel and maintenance. Combining the two means reducing the space available to one of these two functions. For the Allies, there is little need for 'half' an escort carrier; if they're carrying out an amphibious operation, they'll have air superiority from land bases or their large carrier fleets. As such, adding six fighters or four strike aircraft, or some combination of the two, isn't necessary. Landing craft carriers are much more in need, but again, by reducing the amount of craft and troops carried by adding a flight deck, hangar and air group, means that it's not as good as a dedicated landing ship. If you built two of these ships, you'd get a force that was inferior to a force of one escort carrier and one landing ship.