When I was studying Early Music aeons ago, I remember this discussion arising. From what I remember, the tune we now know as "Greensleeves" was originally of Flemish origin and a dance tune. Henry VIII's claim to authorship would thus be about as tenuous as that of Paul Simon to "Scarborough Fair". You'd have had to be a brave man to gainsay him at the time, however, and it does make a good story.
Can't come up with any formal documentary proof at this stage, but it would be interesting to know why the Belgian songwriter Jacques Brel, who wasn't exactly renowned as a fan of "English" folk songs, used a variant of the tune for his song "Port of Amsterdam".
It is possibly the best known early "English" tune, and has been rehashed numerous times over the years. Ralph Vaughan Williams "Fantasy on a Theme of Greensleeves" (which also includes the Norfolk folksong "Lovely Joan") is perhaps the most famous, and there are numerous parodies. There is also the Christmas Carol "What Child is This?" and the 1960s French pop song "Loin" (that's French for "far away", and not anatomical, by the way!).
As Flanders and Swann used to say, "In every period play, set in any time from the Wars of the Roses to the Victorian era, "Greensleeves" is always played - and the royalties go to Royalty!"
But please, always play it in 6/8 and NOT 3/4!!!!