I'd say you're going about this the wrong way!
Even if you did have both GBP and NZD accounts with the same bank, moving money between them isn't "free". All banks I know of offering multi-currency accounts charge a fee or a spread somewhere. Quite a typical rate is about a 3% spread on the currency conversion when moving between the accounts, which is low but non-trivial.
As Gayot pointed out in his answer, it's pretty much impossible to get a regular UK bank to open you an account remotely. You might be able to open a GBP account with your NZ bank, using your New Zealand details, but you'd likely have monthly fees for such an account, plus the spread moving money into it. Some high end (especially offshore) bank accounts might be openable in GBP with only your New Zealand details, but these will certainly have high monthly fees.
Instead, what you should do is just plan to open a regular free GBP account on arrival in the UK. See the How to open up your first UK bank account without proof of address? question for details on how to do that as a new arrival.
Then, before you leave, do two things. Firstly, get yourself a NZ credit/debit card with the lowest overseas fees you can. I don't know the NZ market well enough to know if a "no overseas fee" card is possible (they are in the UK), but get the best you can. You will use this to spend money until your UK account is open. Secondly, open an account with a reputable low fee online currency broker, and setup whatever you need to do to send it money from your NZ account. (If your bank needs to ring you to confirm new online payments, that's much easier to do when in NZ than when you've got to the UK!)
Finally, when your UK account is open, use your currency broker to send + exchange your NZ Dollars in your NZ Account to GBP in your UK account.
In terms of online exchanges / currency brokers, you need to consider a few things. Firstly, reputability and stability - you don't want to wire them money and have them go bust before they exchange + forward! Pick somewhere based in a country with regulations, and someone who's been around a while. Secondly, decide on how often you'll be sending money, and how much. Some brokers charge tiny/no spread on the exchange, but have a fixed fee to send the exchanged money on to the destination bank account. Some charge low spread, but have a variable/% fee on transfers out. Some are poor value and charge high fees for both!
If you're transferring lots rarely, someone with low/no spread on the exchange and moderate onward transfer fees is a good bet. If you're transferring little and often, find someone with % fees which work out less. There's no universal right answer, it'll depend on your situation. (As an example, I've had good service from CurrencyFair, but for small amounts withdrawing cash on a no-FX card when in the destination country has worked out cheaper other times)