Steve Gardham wrote: FWIW I think you have a pretty good choice for origin with Cannibal Islands.

The melody certainly seems to have been widely used in Australia -- and almost all the New Zealand whalers shipped their oil via Australia. Broadsides/songster texts from Australia and New Zealand known to use the tune include

The Settler's Lament (The Beautiful Land of Australia)
The Beautiful Coast of New Zealand (I wonder if that wasn't directly inspired by the preceding)
Going Kangarooing (a Charles R. Thatcher song)
The Indian Empire (another Thatcher piece)
Australian Humbug (yet another)
Dick Briggs from Australia (and another)

If Thatcher used the tune that many times, he must have thought absolutely everyone knew it -- it was easier to sell songs where people knew the tune. Although, of course, Thatcher came along after Weller Brothers was out of New Zealand.