Thanks for all the help, everybody. I loved singing it to myself again.
After a bit of sleuthing, I think the song may have featured in Chaos at Crusty Towers, which was on BBC Music Workshop in 1978.
https://www.broadcastforschools.co.uk/site/File:Music_Workshop_cover_summer_1978.jpg.
Does anyone else think that's right?

I just wish I could find a recording. Such a lovely melody.