My mom, who is from Lancashire, sang us this song while she was trying to hold us still in order to scrub us in the bathtub. This was is S.C. The only verse she sung was,
"Oh Jemima, look at your Uncle Jim -
He's in the bathtub learning how to swim -
First he does the backstroke, them he does the side -
now he's under the water swimming against the tide."
It's a hit. I don't know anything about it, and it may be a few years too late to get any sensible info out of her but I'll ask. Actually her early memory is stronger than the recent stuff.