Lighter wrote: Sammy Kaye and his orchestra recorded "I Used to Work in Chicago" (with "It's a Lie" on the flip side) on RCA Victor 20-2037 in 1946.

(Etc.)

Now this is truly interesting. Notice what we have: a whole bunch of printings of a single verse in the early 1920s, but no recordings. Then, just after World War II, we have a similar explosion, but this time of recordings.

Obviously that says that the piece has a very strong (if perhaps juvenile) appeal -- but it also hints that the single verse is the original, and that someone c. 1946 set the tune and added some lyrics, and that version again became popular. One might conjecture that the poem was converted to a song by Vincent and Skylar, although I'd hate to bet on it.