It looks as though it may have been Edward IV - here's a google search result on this ballad -

ZUR GESCHICHTE DER BLACK-LETTER BROADSIDE BALLAD. Translate this page
by E SCHAUBERT - 1926 - Cited by 1 - Related articles
Broadsides" "Edward IV's wooing of the Fair Maid of London" kurz erwähnte, folgte offenbar Ebsworth, der Eduard IV. in ...
www.reference-global.com/doi/abs/10.1515/angl.1926.1926.50.1

Unfortunately it links to one of those academic sites where the plebs are not allowed to read past the first page, so I cannot check further.

http://www.reference-global.com/doi/abs/10.1515/angl.1926.1926.50.1
However the first page is interesting enough, it tells us that a ballad with a similar title was entered on the stationers' register 1st March 1599/1600. There was a play called "The Fair Maid of London", since lost, performed in 1598 so I did wonder if this song was written for the play.
I agree with Jack's point that old songs may be reprinted for politcal reasons, but I think the 1659 printing is more interesting in this respect - it is a collection composed mainly of songs about monarchs, it was printed the year after Cromwell died, the year Richard Cromwell resigned, the year before the restoration, when lots of power struggles were going on. Some connection?
Thanks for posting this Stower, it has provided more entertainment than a whole book of crossword puzzles!