Nicky, I'm with Beth on this one. I've used my UK bank account to transfer funds to a US account & it worked fine. They charge more the quicker they send it, so if you send it in plenty of time, it doesn't cost as much
Also, I'm pretty sure a NZ bank won't accept a UK cheque.
Have a think about opening a NZ bank account from here, although I'm not sure you'd be able to transfer money from it until you're actually over there & have activated it, but it might be worth looking into. They can open them quickly from here. I'm looking at ASB (who have an office in London) and National Bank.
KG