Greensleeves is not Irish, nor was it written by Henry VII, and it certainly was not by Blondel around 1192.

To put the record straight, it is an example of a passamezzo antico, an Italian minor key ground bass or chord progression, upon which a composer would fashion a tune. The passamezzo antico began in Italy in the 1500s before spreading in popularity through Europe. (The major key version is a passamezzo moderno.) Passamezzo is Italian for 'pace and a half', a term for an old two beats in a bar dance. In the renaissance, the English considered anything Italian to be worth copying. One English lutenist, John Cooper, even styled himself as Giovanni Coprario!   

The earliest reference to Greensleeves specifically is a broadside ballad registered at the London Stationer's Company in 1580 (we can therefore be completely certain that it was not composed by King Henry VIII, who died in 1547), 'A New Northern Dittye of the Lady Greene Sleeves, which reappeared in A Handful of Pleasant Delights, 1584, as A New Courtly Sonnet of the Lady Green Sleeves. To the new tune of Green sleeves.' The tune passed very quickly into English traditional music, being constantly reworked. Our first three records of the written tune are close in date: 'Greenesleeues' in the William Ballet Lute Book, an English hand-written anthology in several hands, c.1595 and c.1610, now in Trinity College, Dublin; 'Green sleeves' in Matthew Holmes' hand-written cittern book, Dd.4.23, c.1595, now in Cambridge University Library; and 'Greene sleves Is al mij Joije' in Het Luitboek van Thysius, c.1595-1620, now in Bibliot heca Thysiana, Leiden, western Netherlands.

By the early 17th century, the tune was well-known enough to be cited three times by William Shakespeare in his 'The Merry Wives of Windsor', c.1602, in which Mistress Ford refers twice to the tune without explanation, and Falstaff later exclaims, "Let the sky rain potatoes! Let it thunder to the tune of Greensleeves!"

As was often the case with ballad tunes, lutenists used Greensleeves as the basis for virtuoso divisions. In the case of Francis Cutting's 'Greenesleeues by maister Cuttinge' in the ms. Add.31392, c.1605, we have a lot more than divisions, with beautiful and unexpected twists and turns.

In 1907, 327 years after its first publication, Cecil Sharp was still collecting morris dance versions of the tune from fiddlers in Gloucestershire. The morris tune 'Bacca Pipes' is a version of Greensleeves and in some morris traditions goes by the latter name. (Two versions, one by each name, can be heard on Volume 9 of Topic's 20 CD set, 'The Voice of the People: Rig-A-Jig-Jig – Dance Music of the South of England').