I heard a version of this from my daughter last year; she was aged 7 at the time. She told me that lots of her friends knew it at her school in the Hutt Valley, New Zealand.
Her words went:
Down by the banks of the handy mandy,
Where the bullfrogs jump from bank to bank,
I said a hip hop
Swaggle waggle hop.
I said a hip hop,
It had a quite complex clapping system between two children to go with it - certainly more complicated than the usual three-part hand clap systems she usually showed me. I videoed her performing this, but can't easily describe it.
Anyway, the rhyme game has made it to New Zealand. Also, nice bit of research Azizi. PM me if you want more details of my daughter's version.