Genie, this can be a long search, leading us ring around the rosy unless Malcolm or Masato pitch in.
The trad Ballad Index cites Gomme, 1898, a "non-ballad play party" song. Found in Britain, Ireland, N. Am.

North Carolina Folklore, Music of the Folksongs, vol. 5, p. 536, has this for "Ring Around the Rosy."

Ring, O ring, O Rosee,
A pocket full of posies;
One, two, three, four,
We'll all tumble down.
Tune "Hoch soll er leben," an old German drinking song.

The "ashes, ashes, we all fall down" is related to the plague of 1347-1350, but this has little support.