Since we're having a qualifications argument here, I will point out that I learned this song in about 1981, from someone who had learned it in New Zealand in the 1970s (very possibly from Neil Colquhoun himself, though there's no way I can test that now), and I started researching "Wellerman" long before it became a hit.

And the typical Reed volume is not a children's books, although they targeted schools. Moreover, they are easy enough to come by that if "Wellerman" were in one of them, someone would have spotted it by now.

The reason I cited the book I did is because it is cited quite a bit in other New Zealand whaling books. Kennard grew up around the Weller settlement, and (IIRC) he kept a record of poetry and such that he heard. Studying everything I could lay my hands on, it appears that by far the likeliest ultimate source for "Wellerman" is the Kennard book.

That is, assuming that (A) there was such a song in the 1840s (dubious), (B) that Tommmy Woods correctly recalled his source after many years (dubious, if one knows anything about human memory), and (C) that all parties are telling the truth.

I don't say "Wellerman" is in The First White Boy Born in Otago; I say that if it actually comes from the Weller period, that that is the place you should start looking.