The National Library of Australia has the sheet music, described thus:

HOME AMONG THE GUMTREES [music] / words and music by Wally Johnson and Bob Brown
[Sydney] : Rosebery, N.S.W. : Mushroom Music ; distributed by Music Sales Australia, [1988], c1975

Other libraries give the following additional information:

First line : I've been around the world a dozen times of may be more [sic].
Chorus first line : Give me a home among the gumtrees.

'The current single from Bullamakanka Mercury 6437150' --cover.
'First released by Captain Rock-Buried treasure L35718' --cover.


There are several videos at YouTube.