I don't think it's incontrovertible at all that "a-tishoo a-tishoo we all fall down" refers to a plague or pestilence. We are dealing with a play-party song (or a children's dancing song for English folks) where "all falling down" is just part of the game. You might as well say "you put your right foot in, you take your right foot out, you put your right foot in and you shake it all about" must be a song about epilepsy!

That's why the plague interpretation is so funny. It takes things that have perfectly easy meanings in that context ("ring around the rosy" or "ring a ring a roses" being an actual ring of children dancing around an actual or imagined rosebush), and gives them new meanings to suit a farfetched interpretation. The line with a-tishoo in it is just as likely to be a nonsense syllable as a sneeze--and anyway, sneezing isn't a symptom of the Black Death! Meanwhile, the things you'd expect to see in a song describing the plague are absent--The actual prominent symptom was not a rash on the cheek, but giant festering boils or buboes under the armpits and elsewhere. Of course, it'd be tough to rhyme that...