I've skimmed most of the Inglis book and found only one reference to an Australian song. A Sydney University songbook, Dirt Cheap, from 1938, had the lyrics of Non-Intervention, to be sung to the tune of Waltzing Matilda. Bits and pieces are given (page 26):

Once a jolly Franco started up a civil war
Liking himself as the top dog you see
....
Aeroplanes from Italy are raining bombs on wrecked Madrid
Gunners from the Volga side are firing merrily
And the League still declares, with the simple faith of infancy
Non-intervention's a reality.

Apparently the song had a go at all the parties to the dispute, but above all, the League of Nations.

Unllikely I'll find anything else in the book, but if I do I'll file another report