It has been quite some time since I found the answer to this out - it took HOURS of almost randomly clicking on links on the Home Office website - but I finally found this page here:
Guidance on how adopted children can become British
This seems to say the process is quite straightforward - BUT is pretty expensive at £1012 (as of Apr 2019)
The fees are listed here: fees for citizenship applications
Plus a fee of £10 for the Biometric enrollment
Plus a fee of £80 for the Citizenship ceremony if the child is over 18 when it is finally approved...
The correct form and it's guide can be found here: MN1
An electronic copy of the form can be found here: eMN1
There is a fairly short list of designated countries this is possible for which can be found here: designated countries
An additional piece of information for those completing this process from within New Zealand. You will need to complete a Biometric Enrollment - and there is a centre available within New Zealand (In Auckland it is in Manukau). Details about how to do that are contained in the UK Visa Application process
EDIT: add in the checklist of required documents You will need to supply the majority of these - prepare early! some of them may take some time to obtain!
The passport issued by The Applicant’s country of Citizenship
Proof of The Applicant’s mother parental responsibility for The Applicant
Proof of The Applicant’s father parental responsibility for The Applicant
Evidence of The Applicant’s relevant adopted parent's claim to British citizenship otherwise than by decent
If The Applicant told us that their parent is settled in the UK then you must provide us evidence of their settled status. This could include a BRP, a letter from the Home Office, an endorsement in their passport or another document certifying permanent residence
Proof of which country The Applicant’s adoptive parents were habitually resident in (meaning the place where they normally live) at the time of the adoption. This can be evidence such as mortgage or rental agreements and employers’ letters.
The Applicant’s birth certificate (showing details of both parents) or certificate of abandonment
A letter of consent from The Applicant’s parent(s) or legal guardian(s) The letter must:
- give their full names and addresses
- say that they agree to let you apply for a British citizenship
If you have more than one parent or legal guardian, they must both sign the letter and confirm that they agree to what it says. If you only have one parent or legal guardian they must explain why the other parent is unable to consent e.g. whereabouts unknown, no contact since MM/YYYY.
A contemporary report from the overseas equivalent of the social services department which details:
- The Applicant’s parentage and history
- The degree of personal contact with the original parent(s)
- The reasons for adoption
- The date, reason and arrangements for the child’s entry into the institution or foster placement
- When, how and why The Applicant came to be offered to the adoptive parent(s)
The Applicant’s parents' marriage certificate
Two referee declarations for The Applicant (one of these should be a person holding a current British Passport)
If The Applicant has a passport then provide this with your application. If The Applicant does not have a passport then you must provide another officially-issued identity document. This could include an official travel document or national identity card
The Applicant’s adoption order
Consent of The Applicant’s adoptive parent(s) to the registration (and in surrogacy cases, a notarised statement of consent from the surrogate mother)
If The Applicant told us that their parent is a British citizen then you must provide us evidence of their citizenship. This could include their British passport, a certificate of citizenship or their birth certificate.
One of the following:
- Where The Applicant’s parents are habitually resident in the UK
provide confirmation from the DfES (for those in England and Wales),
from the Scottish Executive (for those parents in Scotland) or from
the Department of Health Social Security and Public Safety – Northern
Ireland (for those resident in Northern Ireland) that they have been
assessed and approved as eligible to become an adoptive parent. If
there are any doubts about the validity of the documentation
provided, the DfES can be contacted for confirmation that the parents
have had the relevant approval (DfES hold details on all approvals
not just those in England and Wales)
- Where The Applicant’s parents are not habitually resident in the UK provide confirmation from the equivalent of the Social Services Department in their country of residence that all relevant adoption laws have been complied with
Second Edit: Outcome!
I am pleased to report that my daughter is now a British Citizen!