Another possible background to the song would be the Anglican/Puritan conflict in England. The Puritans did not believe in Christmas, as it has no biblical basis; the date of December 25 was in fact selected by the 4th Century Roman Emperor Constantine as Christ's birthday because it had been Mithras's birthday before that. Constantine had a vested interest in converting his armies (largely Mithraists) smoothly to Christianity, which he had selected as his state religion. The English Puritans thus essentially considered Christmas to be Pagan.

Many of the Puritans who came to the colonies--such as the first Governor at Plymouth--expressly forbade their citizens from taking the day off, on the grounds that the economy could not afford people frivolously avoiding work. Anglicans among the Plymouth settlers wanted to celebrate Christmas, but were effectively prevented.

It's purely speculative, of course, but it's possible that this story derives from the conflict between Puritans ("we can't afford NOT to work") and Anglican or even Catholic authority ("you must not work"), with the Anglicans being given the upper hand.

DMcG, I think the concept you were looking for was the Apocryphal Gospels. The Bitter Withy is indeed based on or inspired by a story that was recorded in one of the Apocryphal Gospels. I think it's the one known as Pseudo-Thomas, though it's been a long time since I did the research.