Most of the versions I have are from your side of the pond, Joe.

However there is an interesting article in JEFDSS for 1937 by Annie Gilchrist. There she states the earliest version was in Newell's Games and Songs of American Children first published in 1884.

However she gives 2 British versions dating back to the middle of the 19th century, one from Glasgow and one from Gloucestershire. I doubt it is much older than that.

FWIW there is an interesting possible connection in the Roxburghe Ballads edited by Joe Ebsworth, Vol 8, p 847, in which he quotes a verse from a version of 'Cold, Haily, Rainy Night/Let me in this ae Nicht' from c1830s/40s. The verse runs:

O now so long you've courted me, soldier will you marry me?
O no, o no, that cannot be, for I have wife in my own country.

These lines could have suggested a children's game extension crystalised in Scotland.