My problem with all the "dark and ancient" origin stories is that they constantly lack a middle link. We have on the one hand a possible historical situation that they could be about, and on the other we have a nonsensical rhyme that we hand down to our children or they teach eachother, but no one can ever present the developmental stages between them. We don't have anyone saying in old letter collections or diaries "I heard someone say that ghastly rhyme about the Plague and the Great Fire" either, or something like that, so where were all the rhymes hiding and how were they morphing in between?

My take on nursery rhymes is that they are a great mix of just the joy of playing with words and rhythm, maybe parts of old mnemonics like mayomick suggests, pieces of obscure (to the end users)language and learning (latin grammar tables, parts of the Latin Mass for example) and maybe mythical or historical references.