Green’s Dictionary of Slang

(Aus.) a violent wind storm.

1908		C. Tomalin Venturesome Tom 142: The goldfields after the dust storm, was a sight never to be forgotten [...] the miners rushing to the hotel for drink to wash out, as they said, their eyes and nose and mouth, after this ‘willy willy’ or ‘cock-eyed Bob’ dust storm.
1910		Bulletin (Sydney) 18 Aug. 13/1: I judge that this part of the coast is the most uncertain and dangerous in Australia; that is, in its normal condition, saying nothing of ‘cockeyes’ and ‘willy willies.’.
1926		Advocate (Melbourne) 15 Apr. 29: Such storms as the ‘Cock-Eye-Bob’ and ‘Willy Willy’ are very familiar to the ‘Nor-Wester’ ol West Australia.
1934		A. Russell Tramp-Royal 48: There were no trees there when the Thing had passed. I think we should have fared badly in that ‘willy-willy.’.
1962		S. Gore Down the Golden Mile 5: A willy-willy most likely, a baby whirlwind blowing in from the desert.
1975		R. Macklin Queenslander 106: The loneliness would whip away on the darting swirl of a willy-willy.
2021		C. Hammer Opal Country 19: [H]e can see a willy-willy sweep the mine shafts beyond the airport fence.