I'm so glad you revived this, Robert! I've been searching for decades for a song that was a hit in central Connecticut around 1962 for about a week, then disappeared. From a Mudcat thread on children's chants, I picked up part of it:

atchi katchi boomeratchi (or Liberace).
I love you.

And then on this thread, another part:

kumala kumala kumala vista

I can also recall

5-10-15-20
We don't stop till we get plenty

Googling these got me to this here wonderful article containing multiple variants of all kinds of children's nonsense rhymes and counting games from all over the world.

The article mentions and quotes some commercial songs from 1940s country, 1960s doo wop, and recent rap, all based on those rhymes. But not the one I'm after.Thanks to Mudcat's totally unexpected help, I feel I'm closing in, and I wonder if anyone has ever heard of it.

It was called something like "The Mongolian Cavalry Fight Song," and if anyone knows anything about it, they'd earn my undying gratitude if they shared it.