Well, Bronson has 112 tune variants for Beichan, of which 5 appeared in Greig and Keith's Last Leaves of Traditional Ballads and Ballad Airs (1925).  None of these is obviously MacColl's source, though perhaps three of them might be if we assume that he changed the tune a bit.  Full texts are not given, so no clues there; I must look at the Greig-Duncan books to see if any of the texts match.

The set of Lord Bateman in the DT, incidentally, appears to be a Kentucky version.  Sandy Paton would know it's exact source.

Perhaps the "already here" symbol added to Thomas' post may need to be changed; I don't think that set is already here, and it is Child's primary text, even if somebody has taken it upon themself to alter it.