Your engine with duplicated frame number WM20.113792 comes from military contract S/5209. This was a contract for 12.000 bikes (frame numbers 104818 - 116817). Unfortunately they are beyond the scope of the factory ledgers. But from the Receipt Cards we know that BSA started to deliver contract S/5209 in early September 1944, at 2.000 bikes per month. Which would mean that your engine would have left the factory in December 1944. Here’s a sister bike:



It is also not in the KeyCards, but again, it may have some Greek history as well. After WW2, Greece still had to conquer a civil war. See also here and here.

During the Civil War, the Greek Army (again) received British support. And this support formed the basis of the post war Hellenic Army. Your bike (or at least your engine) may have been used there. These frames / engines are often stamped with a Hellenic Crown with a number. You may find this on your engine as well:

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Just to add to Jan's most comprehensive answer, the British Army units which arrived in Greece from 1944 onwards were frequently those who had come from the Middle East and fought through Italy to Austria. By that time, they had a lot of older equipment and it was generally handed over to the Greek Government when they left. It's quite possible that a 1940 frame arrived in that way.