The tune as printed for 'The Wandering Harper' in Crosby's Irish Musical Repository, linked previously by Karen, has the A and B parts of the tune reversed, as in TTA #2, which is why I didn't recognize it at first.

Looking at some of the other English tunes to Roud 2, none of them is as close a fit as the Adams one, but several are of that ilk. Others, like Harry Cox's, are straight major, rather than Dorian, and it's harder to see more than a vague resemblance there.