I am not familiar with NZ, but pretty much all major carriers I am familiar with have a liaison or electronic version to contact the other airline partner to rebook. Usually after sending a request to the partner liaison, this type of situation can be fixed by Swiss (or affected partner) opening award space to accommodate you. I would try calling NZ and request that their Swiss liaison open award space for you on the flight you want that day, due to Swiss canceling the original flight. Since your flight was operated by an EU carrier and leaves from the EU you should be protected by EC 261 (or whatever it's called) so long as you don't cancel the booking. Swiss has a responsibility to accommodate you and the way to make that happen is to have NZ's liaison contact Swiss to request you be re-booked on your preferred flight, which they should do regardless of current award space so long as there are revenue seats available for sale in the cabin.

Edit:
United, for example, could get this fixed through their Swiss liaison. It may take a few days to process but it can be fixed through their liaison. The key word is "liaison", make sure you use it when requesting NZ fix it. Partner liaison, Star Alliance liaison, Swiss liaison. I'm not sure exactly how NZ refers to it, but liaison is the key word that is important to use when you call. All *A airlines, including NZ, should (must) have a way to liaise with their *A partners to resolve problems like this or they would not be in the Alliance.