I'm guessing since Australia is a republic, they are like OTL India in the Commonwealth, albeit a little more involved with its members.
Not quite.
This is something I struggled with initially, too, presuming that the replacement of a Buckingham-appointed Governor-General with an elected official would result in the reduction of Imperial authority in Sydney. However, even as a republic, the Commonwealth of Australia remains a constituent of the British Empire. Australian taxes continue to flow into the coffers of the Royal Navy and the British Home Island Development Fund, Australian citizens remain first and foremost
British citizens--albeit with Australian residency status--while following laws handed down from the Imperial Council and Board of Trade--the former may be temporary--and the Royal Australian Coast Guard remains at the beck and call to His Majesty, Edward X, as well as the Prime Minister of the United Kingdom and the Lady President of the British Empire.
Whether the Australian head of state is elected or appointed, London expects all British citizens to do their part for King and Country.