doesterr: I wouldn't bet either way & one would think that the writer of of it may have had a specific tune in mind. Shay's "any melody you can adapt to the words" is 1928 which is earlyish even for Bea Lilly or Amsterdam. Also, as aside comment, one of Amsterdam's genius moves was the ability to instantly improvise tunes (and jokes). I think the back-and-forth between adult and childish voices was more important than the tune. Which was for the chorus only.

There is a version out there (I think Dick knows it) which has a tune for the whole song and I think that detracts, not adds. I like Carthy's version & do that.

So there!