With its haunting melody, and the romantic myth that it was written by Henry VIII as a love song for Anne Boleyn, Greensleeves has remained popular over the centuries and, today, is probably the best known of all Tudor songs. There is, however, no proven connection to Henry VIII. Rather, Greensleeves first appeared in September 1580, (some 33 years after his death), and became a hit immediately, with a number of imitations and parodies following.
We’ve explored the history of Greensleeves in this video documentary with Bernadette Banner
We held a conference at the Society of Antiquaries on Greensleeves: Text and Textiles in Early Modern English Culture
For further reading, I can recommend Ian Pittaway's articles on Greensleeves