MLC No.1 for the RN had been completed in 1926.
MLC10 was an improved design first trialled in 1929, and after modifications, again in 1930 through 1934 both at home and in the Med and proved entirely satisfactory. As a result 2 more were bought in 1935 (MLC11 & 12). An improved batch of 6, with 2 engines, were completed early in 1939. Some had waterjet propulsion & some screws. These were used at Narvik in 1940.
Next up was the Assault Landing Craft (ALC renamed the Landing Craft Assault - LCA in early 1942). Prototype ordered Sept 1939, trialled April 1940 with first batch of 17 more ordered Oct 1940. By early 1941 over 200 had been ordered. This craft had a bow ramp to allow the exit of the troops it carried.
Britain placed its first order for the LCP(L) in Oct 1940 - initial batch of 50.
The LCP(L) was followed by the LCP(R) with a "light personnel ramp". The naval architect and author DK Brown, who worked at the Admiralty for many years, credits that as happening after the US entered the war "this idea being borrowed from the LCA".
Info taken from his book
The Design and Construction of British Warships 1939-1945. The Official Record Vol. III: Landing Craft & Auxiliary vessels. Naval Institute Press.
ISBN 1557501629.
In the introduction he notes that this was the publication of an official Admiralty history of ship design drawn up post WW2. He writes
"...This book is published as it was written and no attempt has been made to correct any errors there may be (except obvious spelling mistakes), though manuscript notes suggest that it was very carefully checked."
Now that doesn't quite coincide with the usual timescale of the ramp being tested in May 1941 by Higgins. On the other hand there were a number of USN liaison officers working with the RAF and RN in early 1941 (for example it was an Ensign Smith that was part of the crew of an RAF Catalina and who spotted Bismarck in May and an US Assistant Naval Attache was on Illustrious when she was bombed in Jan 1941, while Britain was training USAAF/USN/USMC officers fighter direction techniques). So it is not beyond the bounds of possibility that information about British landing craft found its way back to the USA in 1941, even if not known about pre-war.
From the LCP(R) was developed the LCVP, with a wider and stronger bow ramp able to accommodate light vehicles.