As others have said, you need to contact the university/ies at which your son is interested in studying.
The outline of the fees situation is that you are treated as a Home/EU student for fees purposes only if you have been resident in the UK/EU for three years previously not simply for educational purposes; or you can prove a continuing and ongoing "connection" to the UK.
This latter condition covers things like Foreign Office amd military families; or families which keep a home (not rented out to tenants) or similar in the UK.
Nationality, country of birth, or citizenship is less relevant than the provable ongoing connection to the UK, or the requirement for residence, not simply for educational purposes. This "connection, but not for education" tends to be in the case of overseas students as postgraduates ie they've done 3 years of a Bachelor's degree in the UK as Overseas students for fees purposes, then apply as Home students for a Masters or PhD. The current regs don't allow that.
On the face of it, from what you've posted, if your whole family looks like it's settled in NZ I have to say why on earth? but I was dragged out to Australia by parents & couldn't wait to get home to England then it's less likely your son would be treated as a Home/EU student for fees purposes. But you'll need to check with universities closer to the time.