No, we are not tied to the UK. Any residual links were terminated in 1986 under the Australia Acts. They only continued from 1926 because we asked for this, with specific exceptions for the states being made to the Statute of Westminster in 1931 which we delayed adopting for eleven years, and then only for leagl rasons during the war.

The reason for the exceptions was not any British wish to control us. It was that the that the state governments, Labor and Coalition, trusted the British more than our Federal politicians. The Queen herself personally played a key role in finding the solution which had eluded generations of Australian politicians.

What we, the Canadians, the New Zealanders and several other countries have is what is well known in international law. This is a personal union, similar to that which once prevailed between the UK and Hanover, and now prevails  between Andorra and France and Andorra and Spain.