The Bardic Museum tune is here: Cornish May Song. A Furry Dance variant.

Gilbert printed the tune only, so we don't know what words, if any, he had heard sung to it.

Roy Palmer (Everyman's Book of English Country Songs, 1979, 222-3) reprints the tune from Chappell's Collection of English National Airs (1838, no.205). Presumably the words, too, are from the same source: though called Helston Furry Dance, the words are a form of Hal an Tow.

If the association of the Furry Dance tune with Hal an Tow is an error, then it appears to go back well before Baring Gould and Songs of the West (which was first printed in 1890, the revised edition appearing in 1905).

Dunstan (Lyver Canow Kernewek, 1929, 26-7) on the other hand, prints the tune first published by Sandys in 1846 (amended), which is more or less the Hal an Tow tune as we know it now, though more complex.

I'm not going to attempt to draw any conclusions from all that. Worth remembering, though, that the Hal an Tow part of the day had died out by the 1880s, and was revived in the 1930s; how much of that revival was based on surviving tradition and how much was taken from antiquarian sources might bear looking into. It appears, for example, that the original dance steps were lost.