Nobody seems to know. Jonathan Lighter has remarked elsewhere:

'No lydian 'Lucy' is mentioned by Bronson.

Nor does Lloyd instance 'LW' in Folk Song in England (1967) in his discussion of the extreme rarity of English lydianism. Martin Carthy's notes to his Byker Hill LP (also 1967) explain:

"A.L. Lloyd, from whom the song was learned, says that in the course of singing it over some thirty years he has emphasised the Lydian starkness of the tune and has also mildly adapted the original (and somewhat scrappy) text."'

He wondered if the tune might have been Irish, as the Lydian mode was more a bit more common there. We don't actually know that it was Lydian when Bert first got hold of it, of course, and in his notes to Skin and Bone (1992) Martin added

'The tune is one of the type that Bert favoured, being cast in one of the very unusual modes. I have not the slightest idea where Bert got it, or indeed if he made it up, but I declare that I don't give a toss, because the feel it generates is, for me, unforgettable...'

On a vaguely related topic (your online cd notes), Bert wasn't responsible for inserting the 'tall and taller' verse into 'The Daemon Lover'. That was Walter Scott, who seems to have made it up (along with another) to fill a perceived gap in the narrative.