What is clear is that any children's song/chant will have many variations as kids add and subtract lines.
I heard it in Kansas about 1950, and many lines were different..including what happened 'on your snout'...it was lice for us.
Also there was an odd 'chorus' after each verse...something like "barumpf-coocoo,barumpf-coocoo,barumpf barumpf, barumpf-coocoo". Maybe that was to allow forgotten lines to surface...

My brother was 3 years younger, and his friends inflicted this on me. I don't think I ever actually 'sang' it.