Thanks All for the response.

AI placed us on Cathay Pacific.

As per agent, he checked with AI and AI informed AirNZ of the flight miss due to bad weather within 3 hours of the flight miss. And as per AI the reservation was cancelled by Air NZ after 3 days. Anyways I dont have anything on paper.

At present AI cant give refund because there is no reservation (return journey cancelled by Air NZ). As per Air NZ the tickets were issued by AI so they are not answerable to me. As per makemy trip they are just a middle man and have little control. makemy trip has not even bothered to reply my emails from last week.
There is no physical AI office in NZ (as per AI site).
It looks like none of them are ready to claim responsibility. I am not sure if I can complain to some governing body maybe IATA? Not sure if they have anything to do with this. But atleast I need to show I have addressed before 26th June the original return journey date. Or else the airlines will come back saying I was late in addressing this problem and hence I cant get a refund.

What about star alliance? can anyone advise.
In the meantime I am checking with AI.