It seems to me; America has a more similar history in common with Australia than the UK.
The similarities between the two nations and social experiments are many; The New South Wales bridge and Brooklyn Bridge, we had Lindberg but Australia had theirs, immigration laws were similar, etc.

Folks in OZ still have an English accent but given 200 more years that will fade. The Great Depression, WWI, and WWII have similar veterans. Australia's fight against Japan and again in Viet Nam and subsequent protests made us more similar.

I could say Chaplin and Australia had their version. Australia opened its arms to Jewish refugees in the late 30s and later in 1947. Women's rights came late to Australia like in the US. along with new immigration laws. We both had a Red scare and a Cold war. We both 'whitewash' our nation's past sins. Racism is similar. We have natives and blacks and you have aborigines. Now we are both multicultural despite conservatives.

We had similar gold rushes. Baseball was American but Cricket was Australia's game. A vintage Cricket game with England became a cultural break with England over their deliberate injury to Australian players. We both are no longer English subjects. Queen Elizabeth rekindled your connection to the failing British Empire. But now what?

YET The US has more ties with English media than Australia. While I know we are united in language, sports, science and music I know very little about Australian current events compared to the UK news. Is it mere distance, the state of Australian media, or just capital?
For now, I thank the internet. Who knows, in another two hundred years of independence, an environmentally ravaged US may envy Australia unless we both burn up. (although the Great Lakes may save us)

Sorry about the American-centric POV and is in no way criticizing the UK.