Oh dear oh dear oh dear ...

Lots of myths without any substance being rehashed here.

1. 'Greensleeves' has no Irish roots and is therefore not a mishearing of anything Irish.
2. It was not written by Henry VIII and has no relationship with any of his wives.
3. The earliest reference to 'Greensleeves' 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.'
4. 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.
5. Greensleeves is an example of a passamezzo antico, an Italian minor key ground bass or common 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. This does NOT mean Greensleeves goes back before 1580 - the ground bass is the chord progression, NOT the tune (just like all blues have a common chord progression but different tunes).
6. No one knows why it is called 'Greensleeves'. There were various livery laws in Tudor England about who could wear what - fabrics, amount of material, colours - so the answer may be something about her social standing. Maybe it just means she's high-born? But you'll need someone who knows a lot more about that than I to comment further - and my guess is they'd still be guessing as to its meaning.