Oh dear, no. See other discussions here and elsewhere of that song. It first appeared in 1939 and, though the tune may perhaps be older, we don't know. It's an easy mistake to make. Disney made it not long ago, and have (I think) settled out of court.

"Skeptic" is right, of course. Folk song studies have always been screwed up by romantics who desperately want to believe that everything is "ancient" or "pre-christian", and insist that all sorts of things must be (though they never seem able to produce an atom of real evidence).

Most of what survives in the West (as opposed to archaeological re-creations) is of no great age (though that's relative: 4 or 5 hundred years is, in historical terms, very little). As "Manitas" pointed out, everybody and his dog from the late 18th century onward was describing things as "ancient" -particularly when in the grip of the newly fashionable Romantic Nationalist movements- that were probably not much older than their mothers.

See in particular the late Bruce Olson's comments in previous discussions here on exactly the same subject. He knew more about all this than any of us. Of (for instance) native Australian tradition I can't speak; but the fact that people have lived somewhere for a very long time is, in itself, no more of a guide to how old a song they sing may be than it is to how long they may have been wearing the same socks.