Hi there
If you want to look at private health insurance, try
Southern Cross Healthcare Group.
However, unlike many countries, it isn't essential to have insurance. The health service here is similar to the UK (some variations, but generally the ballpark cost to you will be the same). And like the UK, non-urgent treatment will take longer than if you went privately.
Some will beg to differ with me - but we've had to use the health service here for both urgent (heart attack) and non-urgent (minor hernia) treatments, and have been very pleasantly surprised. But we do live near Auckland - in more out of the way places there will be differences.