We got a letter delivered to us in teh uk 4 days after the credit to say it was there. National Bank. This would have been reassuring were it not for the fact that we had a phone call to say the same thing the day that the money went in.
4 weeks is loads of time, especially for NZ banks who do the credit overnight, not 3-5 days later like the UK banks.
Given that the exchange rate for cash is likely to be so poor in the UK/NZ (ie if you bring in sterling or NZ$), then you would be as well to just use your UK bank card over here and save the risk/stress of carrying that cash IMHO. Just make sure you tell your UK bank before you come, and your card should workfine.