When the Commonwealth of Australia came into being on the 1st of January 1901 the Commonwealth of Australia Constitution Act, 1900 (UK) defined it "as a self governing colony." (See clause 8). The monarch of the now defunct United Kingdom of Great Britain and Ireland was sovereign. That monarch gave vice-regal authority to an unelected, monarchy appointed Governor-General.
Prior to the end of WWI American President, Woodrow Wilson began pushing for an international body of nations to attempt to prevent any future global bloodbaths. The British at the time still had a sizable empire. The British government decided that if it kept its colonies these would not get a vote in what became the League of Nations. For purely political reasons they decided to grant their colonies freedom, independence and sovereignty knowing that their former colonies would vote with Mother Britain in the League of Nations.
Ninety years ago the British were still the mightiest kingdom on Earth. Their navy surpassed all others. The peoples of Australia, New Zealand and Canada felt safe with the protection of Britain. Australia and New Zealand were on the other side of the world surrounded by non-white foreigners and Canada was living next door to an America already formulating its own colonial ambitions (by any other name).
Truth be told, with the exception of South Africa, Britain's former colonies didn't want their freedom and they didn't want it for very practical military, political, economic and cultural reasons. It was forced upon them for cynical but again, practical reasons.