Well I was always taught that it dated from the time of the Great Fire, and maybe (or I assumed) it was in response to it. But Folksong is about adaptation and mutation, so Folksong definitely. As I remember it was considered a childrens song, but not a nursery song. But that may be because it is the kind of song taught in schools as an example of a round.

And apparently the King dithered about blowing-up houses around the fire storm for several days when he had to sanction the destruction of a greater swathe. He feared compensation to friends and supporters.