The ballad seems to have dropped out of tradition long ago, and no tune is known for it.  The Scandinavian songs such as Ravengaard og Memering are analogues, but we can't tell what their precise relationship to the English story might be (Child and Christophersen notwithstanding); there is no way of knowing if any melodies surviving for them have anything at all in common with whatever Aldingar was sung to.  There is one modern recording of it, by the Breton band Kornog, on their Ar Seizh Avel (On Seven Winds) (Green Linnet, 1985); I have not heard it, and presume that they wrote their own tune or borrowed one from another source.