Not the case.
Australia did not appoint its first overseas Ambassador (Richard Casey, later to become Baron Casey, to the United States) until 1940. Why not? Because until that time, Australia's foreign affairs were still handled from London. And Casey was then, in 1942, appointed (over Australian objections) British Minister Resident in the Middle East by Winston Churchill, exercising Britain's legal right to overrule Australian decisions on foreign affairs. That incident was a main reason why Australia ratified the Statute of Westminster - to prevent it happening again.
Britain didn't exercise its powers over Australia very often between 1901 and 1942, but they were real.
Fellow Australians, it is my melancholy duty to inform you officially that in consequence of a persistence by Germany and her invasion of Poland, Great Britain has declared war upon her and that, as a result, Australia is also at war.
That was automatic in 1939 because of these powers. In Canada where the Statute of Westminster had become active in 1931 (it didn't have to be ratified there), there was actually debate in the Canadian Parliament over the issue and Canada did not declare war automatically when the United Kingdom did, but did so a week later.
"Our King, Mister Speaker, is at war and this Parliament is sitting to decide whether we shall make his cause our own." - The Right Honourable Ernest La Pointe in the House of Commons in the Parliament of Canada.
The Passing of the Statute of Westminster Adoption Act in 1942 was not just a formality. Until it was passed by the Australian Parliament, the law had no force here at all. If it hadn't been for World War II, it is unlikely it would have been adopted as quickly as it was - it was only John Curtin's concerns that Churchill might try and overrule him on issues concerning Australian troops that lead to its adoption in 1942 with effect from the start of the war.
In honor of the late Wing Cammander Smooker, Aussie Airforce, welcome to the trenches.