The first time I ever heard Greensleeves connected with Henry VIII was on a Flanders and Swann record (At the Drop of a Hat back in the late Fifties or early Sixties, and it was part of a comedy routine. Ever since then, I keep hearing people claiming, without qualification, that it was written by Hank the 8th.
There is no proof and no documentation one way or the other. Music historians, musicologists, and scholars specialiing in English Renaissance music all agree that although Greensleeves is often said to have been composed by Henry VIII, there is no real evidence that he did so.
It is conceivable -- just possible -- that he indeed did write it. He was a competent musician (as were most aristocrats at the time) and he dabbled in composing and song-writing.
But what I find baffling is how many people there are who, in the face of such a lack of positive evidence, can be so all-fired sure that he did. I don't know who wrote Greensleeves, and I'm not afraid to say so. The fact that I don't know doesn't keep me awake nights.