President Obama will visit Australia next month.  The trip coincides with the 70th anniversary of US-Australian diplomatic relations.  So, if you do the math, that puts the establishment of relations at 1940.

Prior to 1901, Australia consisted of six self governing British colonies.  Through a process called Federation, the colonies joined together and became the Commonwealth of Australia.  From 1901 to 1940-ish, Australia was in a transition period.  Around the WWII era they became fully self governing, though they still consider the British Monarch to be their head of state.