Doing a quick search on the NZ Herald at Google with the exact line :
site:nzherald.co.nz "john key's government"reveals the NZ Peoples Government being called John Key's government in a number of news articles.
To call a countries government by its prime minister implies that decisions are made by one man. And that one man has greater influence then any other man. Yet that government is for the people and voted in by the people. In my opinion a government should begin with the people. Calling it The New Zealand People's Government has a far better ring to it. A ring which resonates with the people.