I don't think the Polynesians would have had much luck without better crops or animals earlier on.
The other problem is that NZ is pretty far away from everyone. .
There is one theory of early Hemudu exploration circa 7500 BC. The Hemudu were proto-Austronesian group from coastal Yangtze river area.
So Assuming an initial Hemudu population arrives and establishes ongoing communication via Vanuatu/SEA and Polynesia (all of whom are Austronesian and thus familiar to the the descendents of the Hemudu)
The first Polynesian settlement also occured during a warm phase, so that the population was able to acclimatise gradually and the tropical crops had time be established before the frosts wiped them out. If NZ was overall warmer allowing more of the tropical crops (like Breadfruit) to grow it would be awesome and lead to much more rapid ppulation growth. The Hemudu might bring rice or millet too (although Breadfruit has higher yeilds than rice which is why Austronesians changed from Rice to Breadfruit)
You would also have to ensure that the NZs big volcanic eruptions did not wipe out earlier settlements and their gardens too.
Anyway the factors are just right you are going to get a bigger population but probably not in the UK level simply because of distances involved.