A-tishoo is noted in 1883, according to Opie. Two years earlier, Greenway had published Hush! Hush! Hush! Her writings were very popular in both the British Isles and North America and Hush! would be expected there as well. When was Ashes! first noted? You suggest that it is the corruption of A-tishoo, which is possible, but this shift would have more likely taken place in England.
The interpretation of the rose tree and the circle of players as a rash seems a difficult jump; it could have been stated by a popular author or editor of a widely circulated paper or magazine. (Wasn't there a tendency for Late Victorian-Edwardian authors to make illogical jumps like that?). It seems to me that the shift to ashes! from hush or a-tishoo is more likely a "guided" one than one evolved by children.

My (unsupported) suggestion is that there is an author or editor in the woodpile, and we might be able to identify him. Children certainly(?) did not evolve the plague theory.