I've heard about American soldiers needing to return their guns, or buy them back themselves

Neither the British or the Americans, in the 20th Century, were in a situation where anyone reasonably expected to able to keep their issued weapons or to buy them back. In the US The Department of Civilian Marksmanship(which lives on today as the Civilian Marksmanship Program, but that's within our 20 year limit) was historically the vehicle with which surplus rifles and carbines were sold to civilians, often through the NRA. While it might have been theoretically possible to acquire the same rifle you carried in the war through this program, there was/is a bit of a roll of the dice to it.

Now, there was a chance to keep your weapons if you were a member of very specific units of the Canadian military. The Pacific Coast Militia Rangers were issued Winchester 1894s in .30-30. This article on the subject by reddit user /u/othais notes that, at the end of the war, the coastal militia was given the option to buy it from the government. During WW1 the "Montreal Home Guard" was similarly issued lever-actions(in their case Savage Model 99s) and given the chance to purchase them at the end of the war.