So far as I know, the only (published) traditional form with a tune was a cante-fable version from Yorkshire that appeared in the Journal of the Folk-Song Society, vol II, issue 9, 1906, 297-9, as 'One Moonlight Night'. Frank Kidson got it from Kate Thompson of Knaresborough. There was a single sung verse (the 'riddle' one), the rest being prose narrative. Three other versions of the same verse (none with tunes) were appended.

I gather that a (modern) versification of the story later appeared in Kidson's Peasant Songs, but I haven't seen that. If the recording you heard began 'Oh bring with you your dowry, love', that was probably it. Alternatively, somebody else may have put it together; Martin Carthy recorded a form, as 'The Cottage in the Wood', back in 1974; that was a collation from several fragmentary sources, plus some new material of his own. I believe that Deek and Dorothy Elliott may have sung a song called 'The Robber Bridegroom' at some point, but I don't know anything about it.