There was a previous thread discussing this version of All Around My Hat. It looks as if the version we know is a derivative, and the humorous cockney song was the original, as no earlier version is known.

You can also find probably the earliest versions of other songs there. For example Black Velvet Band c1820, set in Barking, London, and William and Diana/Dinia about the same date, which was parodied in Villikins and Dinah.