Perhaps the most well-known disease song of all was Ring Around the Rosy. If I'm not mistaken, it was about the bubonic plague, which created "buboes" on the skin -- red swellings with a ring around them. I think the "pocketful of posey" referred to some mixture of herbs that people would carry in the belief that they would keep the infection away. And the "ashes, ashes, we all fall down" referred to burning the corpses of the recently departed.

I heard this explanation a long time ago, and may not have all the details right. But I'm sure there's someone on this forum who can correct my errors and fill in the missing pieces.