Hi Trevor, thank you for your question. My name is ***** ***** I will try to help you with your questions today.
If the children have their domicile in UK (i.e. having lived there all or most of their life) then your wife would either have to have your permission or seek permission from the Court in UK.
I cannot say that if you refused permission the court would also. If she made an application to the Court to remove the children from the jurisdiction the Court would have to look at what is in the best interests of the children.
You cannot apply for citizenship of New Zealand unless you have lived there for a certain period of time.
If you have any further questions please ask.