For what it's worth, "Australia's on the Wallaby" is in the Digital Tradition at https://mudcat.org/@displaysong.cfm?SongID=388

The first four lines there differ from what Ron Edwards gives in Great Australian Folk Songs. Edwards has

The old man's gone in search of gold,
The claim has proved a duffer,
They search for gold in the rain and cold,
And they are the ones that suffer,

Neither Edwards nor the DT gives any reference that goes back to Lawson's time.