I think that Greensleeves came from into Elizabethan society from somewhere such as Scotland or Ireland. (Queen Elizabeth has one Irish harper in her court that I know of, and perhaps other Celtic musicians were around.) The reason is that there is no other known song in all English literature which is dedicated to anybody's sleeves. In the early twentieth century we find The Blue Skirt Waltz, and that's was close as we come. However, "slieve" is a fairly common word in Gaelic tune names. It means "mountain" or something similar.
If the song came from a conquered people, it is no surprise that its origins remain mysterious.
I used to think that there was no song quite like Greensleeves, but now I have found one. Try playing the beautiful Playford dance called "Daphne" and see what you think.