It shouldn't actually matter what thread form it is , you have the major diameter and the pitch , so anyone with a lathe should be able to turn you up a couple. But it think beezageezer is correct , sae and unf became the same after the war or sometime around then.
The old book of threads (1919) i have doesn't list any details between 3/8" and 9/16" for cycle , suggesting there is a duplicate in some other form , eg. Sae or unf, no 7/16" cycle listed, but 9/16" has 20 tpi so its a good bet 7/16" is 20 threads too.