I don't know how civil registration works in NZ. Would there be a birth registration and adoption registration in the same index. In England & Wales there is a separate adoiption index after 1927
Not for that period.
The original birth certificates are redacted for I think 120 years, or perhaps 125 years. Family members can apply for them. On the index that you can access via the NZ BDM site you will only see the latest registration - so in the case of Kiwi25's mother you would only see the registration from after the adoption which superceded the previous one. This would explain why wivenhoe couldn't see it.
One of my ancestors was registered in NZ as her mother's daughter (no father named) and then soon after as her grandparents' daughter. I have seen both records, but the only one on BDM is the latter one (and that's the one on the Ancestry index). It certainly fools the name grabbers doing their trees (and then to cap it off there was a transcription error with the surname of the man she married, so when I saw that in trees I knew for sure that they didn't know us from Adam)
There is a folio/fiche though available (but not online for the nosey parkers) which just has birthname, region and year (maybe the quarter) that can provide clues.
New Zealand laws must differ considerably from some other jurisdictions. One of my relatives and his wife adopted a boy in Scotland. His biological sister found out in her twenties that she had a brother and was able to narrow down who it could be by going through folios at an office somewhere. She was fortunate in that his adoptive parents had an uncommon surname. I received an e-mail at work in the late 1990s from someone looking for family of so-and-so. It didn't say why, but I had a hunch why. They were able to get in touch.
I met my distant cousin recently. He was very keen to meet me because I was the only person who replied to her e-mail. Most of the people she contacted would not have been from the right family, but none of them replied to say "sorry - we don't have anyone in this family by that name".