The original “Greensleeves” folk song (full name: “A Newe Northen Dittye of ye Ladye Greene Sleves”) was first registered at the London Stationer’s Company in 1580 and is found in many other sources around the turn of the 17th century and is referred to in Shakespeare’s “Merry Wives of Windsor” (written c. 1597; first published in 1602).
It is commonly thought to have written by Henry VIII (possibly due to Shakespeare’s reference) but there is no evidence to support this. Also, the Romanesca chord progression on which it is based was popular in Spain and Italy at the time but is not documented to have been used in England before Henry’s death in 1547.