"I seem to remember reading ( - the old 1-vol. Percy Scholes version?) that Greensleeves was typical example of the Italian tunes imported with the improved viol / ~ modern violin in the 16th century."

I've heard Henry VIII referred to as England's first Renaissance king, due to the fact that he imported new and sophisticated cultural influences from other parts of Europe. The Bassano family were brought over from Venice as musicians and instrument makers to Henry VIII in 1531. If Greensleeves has distinctly Italian overtones, perhaps Henry's Italian court musicians could be responsible? Just a theory, of course, but at least as good as some of the others about grass stains, anyway.

Incidentally, the Bassano family evolved into a noted dynasty of court musicians, also serving at the court of Elizabeth I. The character Bassanio in The Merchant of Venice is thought to be named after one of the family. Another member of the family, Emilia, is the most recent candidate for the "dark lady" of Shakespeare's sonnets.

As it happens, I know direct descendant of the family myself - but he doesn't play the sackbutt, compose Renaissance music, or inspire poetry (that I know of). He's a sheep farmer in Leicestershire.