NZ's biggest exports are meat and dairy products. They import cars, etc. from South Korea. NZ still exports meat and dairy to the UK (and the rest of the EU) and apples, which is useful when they're out of season in the northern hemisphere, but they're not going to start importing cars from the UK, when Japan and South Korea are so much nearer. China is by far NZ's biggest market.
In any case, approximately 80% of the UK's trade is in services and I don't suppose financial traders care where they do business as long as they're making money.
43.3% of the turnout voted in the referendum to replace the existing flag with a Union Jack with another flag, so there's a substantial minority who don't seem to care one way or the other about the Commonwealth.