Well, Joe, it's probably impossible to say for sure. Various tunes have been found with traditional examples of the song, including a major variant of God Rest You Merry Gentlemen with the final line of the refrain of that song substituted for the usual one. The tune best known at the moment (and the earliest known, so far as I can tell from such references as I have) is the one quoted in the Oxford Book. There's more than one tune associated with The Holly and the Ivy, too, so it would be a bit dicey trying to guess what combination or otherwise of tunes might have been used for this broadside version.

Unless anyone comes up with references from contemporary literature (not impossible), we probably can't go beyond speculation.