
It appears to say that the laws of the UK don't apply in Australia, nothing about the Head of State.
The Department of Foreign Affairs specifically states that the Queen is our Head of State.
15.1 Head of State and Governor-General
Australia's Head of State is the Queen of Australia, Her Majesty Queen Elizabeth II. Under the Australian Constitution, the executive power of the Commonwealth is vested in the Queen and is exercised by the Governor-General as the Queen's representative. The Governor-General is appointed by the Queen on the advice of the Prime Minister of Australia. The Prime Minister is Head of Government.
For the record, changing to a Republic has no effect on our membership of the Commonwealth.