As ankirs said, your baby will be a NZ citizen, and also a UK citizen. See here -
Citizenship by birth.
https://www.govt.nz/browse/passports...h-and-descent/
When your register the birth of your child, you will receive his/her birth certificate, and can then apply for a NZ passport in his/her name.
Regarding the British authorities, the information is here.
https://www.gov.uk/register-a-birth and
https://www.gov.uk/get-a-child-passp...child-passport.
As you are planning to travel with the baby, you will have to obtain one or other nationality's passport, but you don't have to have both if you don't want to.
You were granted two years' travel conditions when you became residents. The travel conditions aren't really 'conditions' - it's a name for the permission to re-enter New Zealand as many times as you want to during that two years. After the end-date, you do not have permission to enter NZ again, which is when you need to have got PR to give you that permission for life.
https://www.immigration.govt.nz/new-...anent-resident (Notice, your residence is for ever
while you are in NZ - it's the travel conditions that last two years, then expire.) If you will be back in NZ from your overseas trip BEFORE the conditions expire*, there is no problem. But if your planned trip would last beyond expiry of your travel conditions, you would need to have applied in advance for a Variation of Travel Conditions (see that same link), since you're not allowed to apply for PR early.
*Don't plan to travel very close to the expiry date, because if there is any delay on the journey, even if it is beyond your control (airline problems, bad weather), there is NO leeway allowed. Arrive at the border without permission to enter, and you will be sent back.
Notice, if you travel with your baby on his/her NZ passport, British Immigration at the airport of arrival will be likely to treat him/her at that time as a foreign visitor, even though s/he will have dual nationality - that's not normally a problem because a NZ passport holder has the right to enter the UK for a stay of up to 6 months as a visa waiver traveller, but it has surprised some parents to be asked to show the baby's intention to leave, ticket out of the country, etc..
https://www.mfat.govt.nz/en/countrie...hip-and-visas/