Progression from the League of Nations to the United Nations
The League of Nations was established by the Allied Powers of World War I.
The League's charter was approved as part of the Treaty of Versaillese at the Paris Peace Conference in 1919.
The mission was to promote international co-operation and to achieve international peace and security.
The Treaty entered into force on 10 Jan, 1920, with Australia being an original signatory.
Unfortunately the League was not able to prevent World War II.
The League ceased its work during the war and dissolved on April 18, 1946, and transferred all its assets to the UN which carries on much of its work.
The Charter of the UN was signed on 26 June 1945, and the UN came into existance on 24 Oct 1945.
Australia's date of admission to the United Nations was 1 Nov 1945.

Below are three 'snapshots' from the purported Australian government's own web site:
"Documenting a Democracy" which can be found at http://www.foundingdocs.gov.au/places/cth/cth10.htm
As read from below:
" .... This was a major step in the recognition of Australia as an independent entity in international law. ...
... .... full national status of the Commonwealth of Australia. Article 10 binds each member 'to respect and preserve' the 'territorial integrity and existing political independence' of all other member naions. "

Refer : House of Representatives Hansard 10th September 1919, page 12169.
WM Hughes, Prime Minister and Attorney-General, introducing the Treaty of Peace Dedate,
“Separate and direct representation was at length conceded to Australia and to every other self-governing Dominion.
By this recognition Australia became a nation, and entered into a family of nations on a footing of equality. We had earned that, or, rather, our soldiers had earned it for us. In the achievement of victory they had played their part, and no nation had a better right to be represented than Australia. This representation was vital to us, particularly when we consider that at this world Conference thirty-two nations and over 1,000,000,000 people were directly represented.”
..... See a copy of this further below:

Also, the House of Representatives Hansard 30th September 1921, page 11,631, WM Hughes, Prime Minister and Attorney-General, reporting on the 1921 Imperial Conference:
"We had been a Dominion; the war made us a nation ...." " ... Then came the Peace Conference on which the Dominions were granted separate representation, and sat on a footing of equality with the great nations of the earth."

And, on 11th November 1921
King George V recognised the sovereignty of Australia during the ceremonial acceptance of the credentials of the first Australian High Commissioner to the UK, when he welcomed Sir Joseph Cook as
"… the representative of our ex-colony, the newly independent nation of Australia."