Please allow me to prove beyond a doubt that Greensleeves predates 1580 by at least 300 years.
How many remember the old tv show "Robin Hood"?
One of Robins Merrymen was a minstrel who played a lute.
Robin lived in the days of Richard I (The Lion Heart) 1157-1199 AD
Robin's minstrel played Greensleeves on his lute quite often and sometimes sang the lyrics.
Therefore I would deduct that it must date to at least the late 12th century and is English beyond a doubt. LOL :-}