I honestly don't know, Joe. As I said, this particular rhyme didn't form part of my own childhood.

No: checking back, I find this rhyme doesn't seem to appear in the Opies' work: either the two Nursery-Rhyme books, nor in 'Lore & Language Of Schoolchildren' nor 'Children's Games In Street And Playground'. How amazing. Can't really check with them: tho I knew the Opies -- interviwed them for Folk Review and remained on dropping-in-if-passing and Xmas-card terms for many years, he is now dead and she is 92 and I am no longer really in touch. But, I repeat, I am astounded the rhyme ain't there in any of their works, as I had assumed it must be. Teach me always to be a bit more properly scholarly & check before posting!

Still, its absence from "Children's Games" would appear to indicate, Joe, that the answer to your question as to its being a game-rhyme is most probably that it wasn't.

≈M≈