Thanks,cnd. Although this is not conclusive it is pretty telling.

Another point to make is that it is very much in the realm of children's action rhyme and its popularity means that if it had been in existence before 1939 it surely would have appeared in the many collections that predate 1939. The only entry in my pretty comprehensive children's rhyme/game index with thousands of entries is the one referred to above from my own memory in the 50s.