cnd wrote, I'd wager the post-war proliferation was from soldiers who learned the song during WWII returning home, similar to Mademoiselle from Armentieres in the previous war. The song stuck around college fraternities and similar progenitors of bawdy songs (I saw it in a few college periodicals in the 1920s and 1930s, but none earlier than 1923), where they passed it down orally. Then, in war, soldiers, possibly from similar frats, or possibly just bored, sang them for each other, and then from there it took off.

This is certainly not unreasonable; you are absolutely right that such things happened! And we know that the piece survives in tradition, because of the person on LibraryThing who alluded to the piece in 2021.

The difficulty is that none of the many collections of World War II verse seems to include it; there are no World War II collections in Steve Roud's index, and none of my as-yet-unindexed books of World War II poems have it. That's not proof, but I hate to hypothesize an original when we have an extant alternative.