So New Zealand and Australia (from now on I'll just use ANZ if describing both) we're both clearly western Anglosphere nations during the Cold War who I would expect(although correct me if I'm wrong) would declare war against the Warsaw Pact during a Hot War scenario. Yet they were physically very distant from the main expected conflict zone of Central and Western Europe and were obviously not part of NATO.
As a result, was there any expectation by NATO for ANZ to become involved in defence against a theoretical Soviet assault? Did the Soviet Union have any plans or ideas for dealing with ANZ such as perhaps Submarine deployments to the South Pacific?
Did ANZ themselves have any plans or such in what to do in such a situation? Was the reformation of ANZAC on the cards or expected of either ANZ government? Did ANZ have any expectation themselves to have to deploy to Europe in significant numbers?
Finally, did the progression of the Cold War significantly change the expectations of ANZ and whatever role they might play?
I understand that there are lots of sub-questions and I dont expect them to all be answered. They are there to help give context to the title question.
tl;dr The role of ANZ during the Cold War was to stop the spread of communism throughout Asia, neither nation would've been very useful in an escalation of war due to the most likely outcome of escalation being nuclear and the fact that neither nation is nuclear capable.
Australia and New Zealand weren't a part of NATO and signed their own independent treaty with America called the ANZUS treaty (Australia, New Zealand, United States), the treaty was a three way defensive pact between the nations that was signed in 1951. The treaty saw ANZ become involved in the Korean and Vietnam war. The idea behind ANZUS was to stop the spread of communism through Asia as America already had NATO for Europe, NATO dealt with problems in Europe, ANZUS dealt with problems in Asia. America were more concerned with the spread of Communism rather than an escalation of hostilities wasn't really considered or even worried about. In 1984 there was a disagreement between the US and New Zealand over the movement of nuclear carrying submarines docking in New Zealand and as a result New Zealand left the ANZUS treaty. As for what was expected of ANZ if the Cold War became hot, the outcome would most likely have been an all out nuclear war and the possible destruction of the world. Both NATO and the East Bloc knew that the only outcome of an escalation of hostilities would result in nuclear warfare rather than more traditional boots to ground warfare and Australia and New Zealands role in the Cold War was more to prevent the spread of communism in Asia rather than assist in a potential hot War as neither Australia nor New Zealand were or are nuclear capable.
FURTHER READING
William Tow; ANZUS: Regional versus Global Security in Asia?, International Relations of the Asia-Pacific
Edward W. Gnehm; ANZUS: The alliance through American eyes,Australian Journal of International Affairs