Hi Gina,
This is what is says
here .
Dual Citizenship
As far as New Zealand is concerned there is no restriction on acquiring the citizenship of another country (such as Britain) as well as retaining your New Zealand citizenship. Consequently, you may hold a passport of that other country in addition to your New Zealand passport. Having dual citizenship would not cause you to lose any rights as a New Zealand citizen, although when you return to New Zealand you must use your New Zealand passport or your British one must have a valid permit to enter New Zealand.
So it looks like you could use either but you must get a visa for NZ in your UK passport if you want to use that. Your NZ passport would no doubt prove your citizenship whereas your UK one wouldn't, therefore you'd need a visa in there. If you decide to use your NZ passport, you'd be well advised to sort out the change of name prior to travelling and maybe the NZ embassy could do this for you.
I think most of the more popular airlines i.e. Air NZ, Emirates, Singapore, Malaysia Air etc. have staff that are pretty good with kids but maybe those who have had actual experience can chip in with their comments.