For one thing, there are multiple versions recorded by folklorists and they do not “all fall down.” From the Library of Congress blog:

The claim that the rhyme is related to pestilence is even younger; the folklorists who diligently recorded the rhyme itself in the nineteenth and early twentieth centuries never mention the plague interpretation, although they surely would have had they known it. The first evidence I’ve seen that people were connecting the rhyme with death and disaster is from 1949, when the newspaper The Observer ran a parody of the rhyme beginning “ring-a-ring-o’-geranium, a pocketful of uranium” and referring to the bombing of Hiroshima.

Why mention this? It is just another example of the perils of looking for “ancient survivals” in folklore.