"Lousy" was at least in use in this song by the late 1800's, as collected versions show--they may even show its use earlier. The idiomatic usage goes back at least to the 1730s; the Dictionarium Brittanicum (1736) gives these definitions:

Lousily: in a lousy or despicable manner.
Lousy: infested with lice; also despicable.

In 1867, Archibald Campbell wrote, "Well, gentlemen, you see how he has mauled 'em; you see how he has threshed these lousy, rascally, scabby enemies of yours." A writer to The London Magazine (1784) even indulges in a bit of wordplay involving the literal and idiomatic meanings: "A louse, say the naturalists, is a very lousy animal; and there is not a lousy author in town, especially a dramatic author, that has not fifty lousy critics on his back."

Some commentary on this song suggests that "weary" was a polite substitute for the cruder "lousy", which would indicate that perhaps (like "bloody") it was a word sanitized by collectors and publishers. Where's a lexicographer when you need one?

I'd like to know when the Z sound became prominent--that, to me, is what gives the word a modern tang; but there are some English dialects that frequently voice the S's in words (as evidenced by cant songs), so even that convention may be quite old.