I read the same web site mentioned above, and also went down a rabbit hole. I did, however find another tune titled Follow the Drum, which might actually be the song referenced in Humphrey’s book (see Mainly Norfolk) as mentioned in the website. I believe the website’s author mixed up the modern song with the older song. The song sung on the show is by Rael Jones, as also mentioned above.

https://mainlynorfolk.info/folk/songs/thesoundofthedrum.html