Years ago I read in an academic book on folk songs and ballads (can't remember which one, but I probably have it in my bookshelves somewhere) that the first known printed copy (a broadsheet) of the song bore the notation:Newe words to the olde tune of 'My Ladye Greenfleeves'
The new words were the ones we probably know the best: Alas, my love, you do me wrong . . . etc., and it was dated 1580.
The book also said that although the melody is sometimes attributed to Hank 8, it is very unlikely, and that the best one can say is that its origins are unknown.
Bit of trivia: my wife,Barbara, who owns an old letter-press (weighs about a ton and is stored in a friend's studio) and knows volumes about printing and its history, informs me that the reason the s sometimes looks like an f (minus) the crossbar) is that when it's next to a taller letter like an l, it helps to support the adjacent taller letter. Blocks of type were often made of wood, and they broke easily. That also accounts for some weird conventions in punctuation that still survive.
As a friend of mine used to say, "Greenfleeves. What a ftrange name for a fong!"
Don Firth