For the song that is burlesqued here check out a contemporary broadside 'The Green Willow, or All Round My Hat'. There are at least a couple of versions in the Bodl and quite a lot more in Madden. None of them appear to be any older than Pitts/Catnach, but as the trend of the period was taking serious ballads and burlesquing them the serious one is very likely the older, but not by much.