There's no evidence that Henry even knew the tune, and the attribution to him is later, and apocryphal. There is no reason to think that it was "traditional" at that time. The earliest reference to the title is in the Stationers Register (3 September 1580), and it's likely enough that the tune was composed around then, though it didn't appear in print or surviving MSS until considerably later.

What Child is This, far from being "Elizabethan", is a 19th century text, written by William Chatterton Dix. It was later set to the Greensleeves tune in American hymnals.

All this has been gone over in long and tedious detail in previous discussions. There are very few songs about which more nonsense has been put about than Greensleeves, I'm afraid; and most of it can be found here by those with the patience to plough through it.