Malcolm, I think it probable that the Scandinavian and English-language songs are directly related. The story and the motifs are so similar.

The Gaelic song which some people link to Child 10, on the other hand, is quite different. The women are unrelated and one is already married with children, also there is nothing in the song or related narratives about the dead woman's body being found and made into a musical instrument.

As for the tunes, even within any one language group there are versions of the song with similar lyrics but disimilar melodies.