My opinion, if your kids have been receiveing immunisation in NZ, you can just take them to immunisation, i think the nurses there will give you some ideas if you are using NZ Child Health Record, before that, please make sure the immunisation session schedules
And you can ask your local council about the address of local Maternal and Child Health Centre, Maternal and Child Health Centre may able to issue a new Child Health Record book, if they are not, they still can give you a hand.