I've been trying to sort out where the tune comes from and not having much luck.Alice theorized that Milligan wrote it on the basis of the credits on the John McCormack album, but I haven't been able to substantiate this. The record I have just says "Milligan-Clay" and doesn't mention composer or arranger like the info from the McCormack website.
It sounds like Milligan may have collected the song or at least arranged it. Perhaps McCormack used Milligan's arrangement that Clay then tweaked in his accompaniment, thus the credit to Clay as well. I guess the tune and love song lyrics are both traditional, though I'm surprised that McCormack didn't identify it as such. If anyone has further info, I'd be very interested to hear it.
Zena