I've always thought of it as Appalachian, and was surprised that it was featured on British radio.

I just did a bit of checking and it is in the Jean Ritchie songbook (Folk Songs of the Southern Appalachians as sung by...), though not from her family tradition. She heard it sung at the John C. Campbell Folk School in Brasstown NC, and says it is in 'Songs Of All Time', a little book of folksongs from all over published by the Cooperative Recreation Services of Delaware OH.

It's not in my copy, however, which is a 2nd edition from 1957. (First edition was 1946.)