Hi Bstar,
There's a form to fill in - I think it's a P85
HM Revenue & Customs: P85. They will decide whether you're still a UK tax resident or not. You might get a tax refund, although they usually take ages to process, so you're more likely to get a bill for overpaid tax credits......
You have to register with the IRD in NZ for an IRD number before (if) you start work. You can print the form off here -
IRD number application - individual IR595. It takes about 10 working days to process, but you need a fixed NZ address to apply for it, not a post box.
Best of luck.