Much of this is repetition of thread 9724 (the first related thread linked above). The South Riding site, with credit to Humphries, was posted by Malcolm Douglas, 05 Jul-02, with the words in two posts by Dicho (Bodleian Collection with added verse from South Riding text).

Amos, thanks for the kiddies versions. There must have been a lot of these since the song's appearance well over 150 years ago, and its use on the minstrel circuits as well.
I wonder if there are any more short playground rhymes. The potatoes verses posted by Amos surely must be one.