For what it's worth, I looked up what Ed Cray had to say about this in The Erotic Muse. He learned it himself in high school in 1948 -- but, of course, that's after the recording explosion, so that's not proof of anything. He mentions some analogous songs like those mentioned by Bruce O. above.

His tune has the same opening phrases as "Malbrouck"/"We Won't Go Home Until Morning"/"The Bear Went Over the Mountain"/"Pig in the Parlor," although the final phrases are different. He observes that line that is similar to "The Bowery" also has a tune that is similar to that part of "The Bowery," and suspects that the "Malbrouck" tune was attracted to the latter.

He mentions that Kenneth J. Larson's unpublished 1952 article "Barnyard Folklore of Southeastern Idaho," which includes collections going back to the early 1930s, included a version of the piece.

It's in the 1971 edition of Immortalia, but not in the original edition.

Oscar Brand wrote to Cray in 1971 saying that he added to the piece and claiming credit for its popularity based on his 1951 version. Given the list of recordings before that, I think we can disallow that claim, though Brand probably did add verses.