Thought I'd add to this discussion because the song got mentioned in the current 'Filthy Songs' thread. Both the 'balls' and 'boots' version have been known here in New Zealand, from at least the 1940s.
Maybe there is a relationship between this song and 'A Lovely Bunch of Coconuts'. There are a few similarities in the chorus:
Oh, I've got a lovely bunch of coconuts,
There they are all standing in a row.
Big ones, small ones, some as big as your head
Give 'em a twist, a flick of the wrist,
That's what the showman said.
For they were large balls, balls as heavy as lead,
With a flick and a twist of his muscular wrist,
He could throw 'em right over his head.
Or maybe it's entirely coincidental...