As far as I know, the Cuckoo Waltz is completely different from any of the folk songs referring to cuckoos.
It was the theme tune to a sitcom of the same name, set in Manchester, in the 60's or early 70's, and didn't have any lyrics. I can hear the tune in my head now, but have not the technology or skill to transcribe it, sorry.
Cheers, Bron.