Malcom - thanks for that. I can now see where the cow in the Adderbury song may have come from. There was no hint in the other versions. I had looked at them already, but couldn't even see how they fitted the tune, so I thought it may not be connected.
Other notes: Some years ago, when working in the area, I attended practices of the Bucknell (ladies) side, and learned their version, called Old Black Joe. This makes a much nicer tune to play on the D/G melodeon than the Adderbury one, especially played across the rows to get a more interesting chordal accompaniment. Also, I play a tune learned from a Callenig recording, called Galch Gwyn Morgannwg - White Lime of Glamorgan. This is a version of the White Joke.