I agree with the above. Unfortunately for the OP, their contract was to get him for BHX to AKL which they did. They are not responsible for any on-carriage that you may have.

For anyone who may be interested, NZ do offer 'HFL' fares which are combinable with any airline's fares. They are fairly expensive but should have been able to be added to the OP's ticket. Once an HFL is added to the ticket, that sector (in this case would be AKL-DUD) takes on the same fare rules as the international sectors.