Seems like the original purpose of this thread is done and I hate to make it longer. Not enough not to do that though!

The ‘down by the banks of the hanky panky’ version I know is very similar to a lot of the shorter one-verse versions here. I was taught (by a teacher or similar) a song with the same tune and shape, to the same try-and-avoid-the-hand-slap circle game, that went:
Down in the jungle where nobody goes
There’s a big fat gorilla, blowing his nose...
[then some sounds that are supposed to be like him blowing his nose. I don’t remember them, sorry!]

I grew up in London, England; my friend who grew up in oxford & cambridge knows the gorilla words but not the frog words. In my school we played ‘oranges and lemons’ more often in the playground - hanky panky was more of a sitting down game.