Re the suggestion from Young Buchan. I found your post when googling a rhyme which had passed down through my family over the generations. Where the version you quote goes:

Hy me rinkum cokum candy rip tay Johnny with a rum tum day
Hoker flip flap jump Jim Dandy rang hi Shanghi shum tum alay.

The version I learnt from my grandfather is:

Oh my rinkum cokum candy up comes Johnny with a rum tee flo
Hop skip flip flap jum gin jandy twanguy tawnguy chum chum baloo.

The rhythm and the many of the words are the same so one obviously derives from the other. Do you know anything else about its history?