I don't think a tune was ever documented for the Craven Churn-Supper Song. Curiously, when that radio feature was made about 'Songs from the Ridings' (see http://www.eyes-and-ears.co.uk/pennine/details.asp?Title=Three%20Nights%20Drunk%20and%20the%20Market%20Tup) this song was mentioned in the blurb, despite not being in the book. I was intrigued about this and asked Nigel Hudleston about it, but he didn't know the song. I even got in touch with the show's producer and asked about it, but she hadn't a clue.

For what it's worth, the tune of Wir pflügen und wir streuen (We plough the fields and scatter) fits this one quite neatly. Seeing as plenty of German tunes found their way into English singing, it's not too incongruous!