Have you seem this document (or is this your source for the Exton song?): Elizabeth Ruddock: May Day Songs and Celebrations in Leicestershire and Rutland.

At a quick look, there's a convoluted section about parts of a song spread around several villages and the Exton song is amongst that section. I'd need to read it a bit more thoroughly to tell if the Exton song is meant to be sung to one of the two tunes mentioned in that section. If a closer read doesn't resolve it, try contacting the author.

Mick