There is a published song called WHERE THE DOG SITS ON THE TUCKER BOX with words by "Alf" and music by Jack O'Hagan, c1938.

The sheet music is included in a book called "Soldier Songs for Camp and Canteen" published in Melbourne by Allan's Music Australia, c1941. You can see online images at The National Library of Australia. These are the only lyrics given there, but I suspect there are more:

My Mabel waits for me underneath the bright blue sky,
Where the dog sits on the tucker box five miles from Gundagai.
I meet her ev'ry day and I know she's dinky di,
Where the dog sits on the tucker box five miles from Gundagai.
I think she's bonzer and she reckons I'm good o.
She's such a trimmer that I've entered her for the local show;
And my Mabel waits for me underneath the bright blue sky,
Where the dog sits on the tucker box five miles from Gundagai.