The point is that Air India need to contact the Air NZ Agency Sales department on your behalf, for the seats to be reinstated.

I assume it was Air India who rebooked you on CX - if so they failed to correctly cancel the HKG-AKL NZ sector, so in Air NZ's eyes you failed to show for the trip and they cancelled the rest of it.

An Air India-issued ticket, that was rebooked by Air India staff, needs to be dealt with by Air India. The AKL-based Air India office needs to go through Air NZ Agency Sales on your behalf, it is entirely their responsibility not yours, and you need to pressure them to do it asap.