Does it matter? I've always thought of Greensleeves as an English song. The tune and form certainly sound more English in their presentation than Irish.

But an awful lot of tunes that might be passionately claimed by one country are equally passionately regarded as "mine" by another territory! Look at all the Irish tunes that those damn Eastern Europeans claim as their own, and vice versa!

Play the music and love it for what it is!