The piece left out of the puzzle in the above posts is that the plague interpretation is itself not old. There is no old folk knowledge about a connection between the plague and this rhyme. The claim seems to have been made only in the mid twentieth century, if Philip Hiscock of memorial University Newfoundland is correct.
In the absence of any sort of old folk knowledge, we can reasonably ask what evidence the mid twentieth century originators of the plague idea had. First of all, the interpretation is based essentially on one variant instead of on the huge range of texts that exist. Second, it proposes meanings for some of the words (ashes referring to cremation, ring around the rosie referring to a rash on the cheek) which are clever but not compelling. Third, even if they were compelling, why a medieval plague and not a nineteenth century outbreak of some disease?
All this suggests to a folklorist (which I am) that the whole thing is a red herring. You have to assume that the one text being analyzed retained all the oldest features (for which there is no evidence), that this text contains cryptic refernces to disease (for which there is no evidence) and that this disease was a very old plague (for which there is no evidence). In the end, whether a folklorist believes a given interpretation or not is based on evidence, and there is just no evidence of any kind that this interpretation is correct, besides a clever correspondence between some meanings of some of the words and some of the features of some diseases.