As a Kiwi, I've been taught a lot about the Treaty and English settlement of New Zealand, but for some reason it sort of skips from Europeans arrive; to NZ is part of NSW; to Treaty being signed and then we learn from there.
Did something happen in between? Did New Zealand separating influence the Treaty at all? Was it simply a matter of distance/travel time/simplicity?
New Zealand was basically an independent colony even when it was officially part of New South Wales - the Lieutenant Governor made decisions autonomously without ever consulting his supposed superior in Sydney. The only real significance of New Zealand being part of New South Wales was that New South Welsh law applied. The intention had always been to separate New Zealand into its own colony - the New South Welsh connection was a temporary stopgap.
It didn't effect the Treaty because the Treaty simply stipulates that the British monarch is sovereign - it doesn't stipulate any particular arrangement for that sovereignty (e.g. both NZ being part of a larger colony and NZ being an autonomous colony satisfy it).