Australia isn't part of the UK. It's been a separate country for over a century now. However, Queen Elizabeth is Queen of Australia.
Incidentally, she's also Queen of Canada, Queen of the United Kingdom of Great Britain and Northern Ireland and its Dominions across the Seas, or whatever it is now (it used to include Emperor/Empress of India). In short, Queen of a bunch of places. Lucky her.
Australians, despite their self-appointed reputation for being irreverent and contravening authority, are in fact deeply conservative. They follow the rules (and the wheels). They are suspicious of change: any change. I think they operate on the 'if it ain't broke' principle.
But the links to the UK are more cultural than anything else these days. Australian humour, language, etc is still mostly interchangeable for its source in the British Isles.