An obvious point is that there are no "sixteenth century English folk songs" -- the term hadn't been invented yet. There are, of course, songs which were in oral tradition in the sixteenth century, a (tiny) handful of which were still around in the nineteenth and twentieth centuries for collectors to find them. Although it is very rare for them to use the same words.

There might, to be sure, be a sixteenth century song of unknown origin which is the ancestor of this thing. But, if so, it has clearly been modified.

And I certainly have no record of anything like this in any recent anthology. My strong suspicion is that someone found a sixteenth century poem, modernized the bleep out of it, and called it a "folk song." And then, as too often happens, others copied the attribution.

I can't prove it, of course. It's just the old rule that "it is much more likely that a modern composer was lying than that a song survived unmodified, yet somehow in twentieth century English, in the sixteenth century." :-)