Thanks, Jerry.There being no evidence for something before 1580 simply means that it's at least as old as that, not the other way round.
I'm not convinced Henry wrote the song - there's no evidence. But that means there's currently no evidence he didn't either. I'm fairly sure that he did, however, write a number of other songs and poems as this is fairly well established. See this site for useful info.
Cheers!
Ian