Haven't we talked about this before?
I remember a similar game played at children's parties (organised by adults) in London in the 40s.
Called 'Tippit' you stood in a circle holding a string loop with a ring on it. Everyone pretended to pass it on, the person in the middle had to guess where it was, when they found it the person holding it took their place.
No song or rhyme that I remember.
And the person in the middle could tap any hand they chose, and see if the ring was there.