They have unique accents in their spoken English and have some amusing words they have added. However, generally, their formal written English is UK English. My understanding is that many linguists would argue that UK English, American English and Canadian English are the distinct forms of English with most other English speaking countries speaking UK English.
Personally, I have often wondered how they would classify some Jamaican English as it really does not seem like any other English to me. However, having lived in NZ and having spent a few months in England, I have to agree with them that NZ English is pretty much UK English with some amusing terms added.