As always, child lore proves startling and confounding of expectations. Up till I started this thread I was so so confident about this song being almost exclusively Scots. Thanks to all who proved me wrong, and gave me a new interesting puzzle - if it was so widespread in the US of A, why does it turn up so rarely in print? (Or am I missing things here too?)
Re Jimmy Rushing - at that time there was a fashion for jazzers to 'jazz' up kids songs. Roy Eldridge did a medley called Schooldays. The best known was Mairsy Doats and Dozy Doats, by I do believe Nat King Cole. This is very old, as 'Mares eat oats and does eat oats'.