You may have misunderstood a little, Alan. The title is as Masato gave it (apart from the typo!); John Swabber is just a short part of the book (the first edition, published -conjecturally- in the previous year, was of 40 pages only) which is a collection of short theatrical pieces. The author, Robert Cox, is described as a "Comedian"; that is, a performer on, or writer for, the stage. There seems no particular reason to suppose that he didn't write the words of the Maypole song, though I've seen it suggested that Singing Simpkin, also included, may have been based on one of Will Kemp's routines. The copy of the second edition held at the British Library is of 13 pages, and contains only an additional piece, Simpleton the Smith.

The Maypole tune indicated for that broadside would perhaps be Joan to the Maypole ("Come lasses and lads").