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.

In Oz (and I'm sure we didn't originate it) there is a Morris tune that is known as "The bacca pipes jig" which is "Greensleeves" played in a sprightly 6/8.

And, as Tangledwood and Rich-Joy have both mentioned, Greensleeves as the theme tune for Mr Whippy vans has been out here since the early 60s. In fact, a friend of mine from student folk scene days got his solicitor's practice ("attorney" for those from the US) up and running partly by pursuing icecream vendors who "infringed Mr Whippy's copyright on the use of Greensleeves as the tune for soliciting customers". Given its age (as presented by Mr Red, above) I suspect the only leg Mr Whippy had to stand on in such an assertion would be "ownership" of their "arrangement" (thus dignifying their contribution well beyond the bounds of taste, if not credibility) of the tune.

Cheers, Rowan