Peterr: I'm speaking from memory and I've lost a few books over the years.Possibly Baring-Gould's "Songs of the West Country"(at a guess) married the words to the tune,probably in 1900give or take a year or two. Being in a published book influenced a number of performers, both in the more classical world and in various folkie revival circles, performed it that way.
They do sort of fit: try it
Robin Hood and Little John
Are all gone to the fair-O
And we will to the merry greenwood
To see what they do there-o
(That is the version in the book I remember). It may be ,of course, that these words actually were sung at some stage to the Furry Dance tune, but I believe this to have been an error by Baring-Gould or his informant. Someone else will know more about this than me, I hope elucidation will arrive from somewhere.