It seems to me that since WWII, most of the Anglosphere have acted almost identically diplomatically. They also all share a language and have similar cultures. It seems to me it would be pretty wise for them to join up and form a sort of consensual empire.
The states you're talking about (Australia, Canada, New Zealand, UK, USA) have been joined militarily since WWII through the UKUSA Agreement and the ACBA armies program. They also constitute the members of Five Eyes, possibly the most significant intelligence cooperation agreement in operation today.
The UK and USA have their "special relationship" and Australia and New Zealand have aligned strongly with the US in every major conflict since WWII. The ANZUS Treaty commemorates this Pacific Anglophonic union. Britain maintains strong ties to Canada, Australia and New Zealand through the Commonwealth and the position of the Queen as the head of state.
Essentially, there is no desire or need for supernational or supranational union above the alliances and agreements already in place.