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There's not really a coldest place in NZ, except like in the Mountains.
But The lowest temperature ever recorded in New Zealand was —22°C at Ophir, Central Otago.
When you visit go for the South Island, it's full of mountains - the Southern Alps, so places nearby get pretty cold, the West Coast, Otago, Southland and even Canterbury can be quite cold in winter, which is June-August.
In the North Island Rotorua and Taupo would be the more cold areas, and the nearby towns to Mount Ruapahu. Even where i live - Wellington can be coldish aswell.
But not as cold as in the South.. In the winter it won't really get over 15 C.
It will generally will be below 10 C and can reach negatives, in Otago, Southland...on the mounts it should easily get into negatives - specialy overnight it will be quite chilly. The nearby towns and cities to the mounts can get snow, but it's not common, just a once a year kind of thing if not at all.