The Song Greensleeves has been documented to about 50 years after Henry VIII's death. The inference is that he might have written it but the evidence is not there. He was an accomplished musician. There is a record of a Morris tune called Greensleeves (from the time of his encumbancy) but I am not sure if we know how the tune went.
Here endeth the gospel according to Paul Burgess (who is erudite on these matters).

I have no doubt the street sellers of old were regarded with less than universal cuteness. Plus ca Change.