I'd certainly quarrel with Greg's argument (particularly the analogy!) though in essence it's a perfectly fair point, in answer to which I can only plead tiredness; I should have been more specific, and have paid for it now! Though a form of the rhyme is on record as having existed at the end of the 18th century, it appears at that time to have been a simple dancing game:

Ring a ring a rosie,
A bottle full of posie,
All the girls in our town,
Ring for little Josie

The Opies further comment

"The A-tishoo is notably absent here, as it is also in other versions [William Wells Newell, Games and Songs of American Children, 1883] gives, in which the players squat or stoop rather than fall down:

Round the ring of roses,
Pots full of posies,
The one who stoops last
Shall tell whom she loves best.

They go on to dismiss fairly comprehensively the "Black Death" myth, which appears to be a fanciful invention of the 20th century. I think I've quoted them on the subject here before, so there's no need to repeat it now. As I've said, the book is easily available. All of this doesn't prove, as Greg rightly points out, that sneezes and falling down did not occur in earlier versions; since, however, their first known appearance is of some 90 years after the first recorded forms of the rhyme, the balance of probability is that they are, as I said (though I should have suggested) later accretions. Whatever, there is no evidence of any kind that the rhyme is older than the 18th century, unless further material has been unearthed since the Opies wrote on the subject.