from Australia's War 1939 - 1945:
On 3 September 1939, Prime Minister Robert Gordon Menzies announced that Australia was at war with Germany:
"Fellow Australians, it is my melancholy duty to inform you officially,
that in consequence of a persistence by Germany in her invasion of
Poland, Great Britain has declared war upon her and that, as a result,
Australia is also at war. No harder task can fall to the lot of a
democratic leader than to make such an announcement."
From speech made by Prime Minister Robert Gordon Menzies, 3 September 1939
A public announcement by one's duly elected Prime Minister seems definitive to me.
Australia's Statute of Westminster Adoption Act 1942 explicitly states that five provisions of the Westminster Act (sections two, three, four, five, and six)
are adopted and the adoption shall have effect from the third day of September, One thousand nine hundred and thirty-nine.
Sections three and four of the Westminster Act bear on the question (my emphasis):
3. It is hereby declared and enacted that the Parliament of a Dominion has full power to make laws having extra-territorial operation.
4. No Act of Parliament of the United Kingdom passed after the commencement of this Act shall extend, or be deemed to extend, to a Dominion as part of the law of that Dominion, unless it is expressly declared in that Act that that Dominion has requested, and consented to, the enactment thereof.
- that the UK Declaration of War on Germany passed the House of Commons
on Sep. 3, 1939; and
- said date of passage is not after the commencement of this Act [effective date of Statute of Westminster 1931, being Sep. 3, 1939, as regards the Commonwealth of Australia],
it seems clear that the date of effect was deliberately chosen to ensure that Australia was legally in a formal state of declared war against Germany prior to the Westminster Adoption Act taking effect.
Update #2 This interpretation is reinforced by the signed declaration beneath the Act by the Clerk of the House of Representatives (my emphasis again):
I hereby certify that the above is a fair print of the Bill intituled "An Act to ... by adopting certain Sections of the Statute of Westminster 1931, as from the Commencement of the War between His Majesty the King and Germany", which has been passed ....
This combines to ensure that Menzie's broadcast declaration above (of Australia being at war with Nazi Germany) was not impeached by the retro-active application of the Westminster Adoption Act 1942.