This page
https://www.gov.uk/apply-citizenship-british-parent may help. It occurs to me that when you enter the UK, if your children are UK citizens and hold UK passports, like you, passing through the formalities at the airport upon arrival will be easier.
This
https://www.gov.uk/register-a-birth is another procedure you can go through while still in NZ, which would mean that any UK official checking either child's identity would find him/her in the British records.
This page
https://www.gov.uk/uk-family-visa/partner-spouse, then later, this one
https://www.gov.uk/government/public...e-uk-form-setm, may be useful for your partner.