] Would JA have sung carols? Or known them?

I'm curious about my favourite carol. I know "What Child is This" is set to the tune of "Greensleeves", and "Greensleeves" goes back to at least the time of Henry VIII (?). I say Henry VIII because I remember hearing somewhere that he was rumoured to be the composer of "Greensleeves".

So when did someone set Christmas words to the tune?