William Chappell wrote, in Popular Music of the Olden Time (vol. 1, 1859, pp. 227-8; underline added):

The earliest mention of the ballad of Green Sleeves in the Registers of the Stationers' Company is in September, 1580, when Richard Jones had licensed to him, "A new Northern Dittye of the Lady Greene Sleeves." The date of the entry, however, is not always the date of the ballad; and this had evidently attained some popularity before that time, because on the same day Edward White had a license to print, "A ballad, being the Ladie Greene Sleeves Answere to Donkyn his frende." Also Edward Guilpin in his Skialethia, or a Shadow of Truth<>I>, 1598, says: "Yet like th' olde ballad of the Lord of Lorne,
                      Whose last line in King Harries days was borne."
As the ballad of The Lord of Lorne and the False Steward, which was entered on the 6th October, 1580, was sung to the tune of Green Sleeves, it would appear that Green Sleeves must be a tune of Henry's reign. Copies of The Lord of Lorne are in the Pepys Collection (i. 494), and the Roxburghe (i. 222).

Who originated Henry VIII's authorship?

~Masato