Can I watch movies I’ve purchased in the Apple TV store anywhere in the world?
Not “anywhere” in a literal sense, but NZ-to-CA might actually work for some (or many) titles.
The Apple Media Services terms and conditions [NZ] [CA] state that services are available to you in your ‘home country’ of that moment, meaning no guarantees for different countries, nor for switching countries.
When you change the country for your Apple ID account, then you get access to a different iTunes Store region, which will have a (partially) different movie catalog (with potentially different language support). Movie titles that are considered a matching Store item in both regions, may still be available to you after the switch. This might be the advantage in your situation, considering the cultural and language overlap.
An officially supported way to get all your movies over to the new country, is to download the movies on Mac (or PC). The downloaded copies get authorized as your purchases, even when not available in the new country, but you would have to stream locally (on the Mac or Home Sharing), as online streaming would try to access the new region’s iTunes Store instead of the downloaded copy.
Change your Apple ID country or region - Apple Support