We had British passports and British currency. There was a lot of nationalistic work put in after World War 2 so that Australia and NZ would see themselves as completely separate countries.
No we had Australia currency. There certainly was a breakaway from Britain during WWII after Churchill abandoned us and the USA came to our rescue. Curtin pulling the Australian troops out of North Africa was probably the point of no return.
It was a slow divorce from the empire, but it was well advance pre WWII.
In some ways the relationship of Australia (and NZ) was similar to that of the countries of the EU. There was free movement of people (provided you were white) and free trade around the empire, but most people thought of themselves as citizens of their own countries.