British Expats - Health care

Since you are on a visa that allows you to live and work in NZ for at least 2 years you and the family will be treated as Residents / Citizens so you'll get subsidized care when visiting the GP.
Unlikely your job healthcare will cover all the costs for you and the Mrs....likely you will have to pay the subsidized cost upfront and you then claim 80% or so back through your healthcare provider up to your policy limits OR you use your healthcare provider card and just pay the difference at the GP's after you have had the consultation/treatment so long as they accept the card of course.
Kids are zero fees for under 13's but you'll pay for repeat scripts and if they are faxed straight to the pharmacy etc.
Visiting emergency dept is covered. Referred treatment is covered. If you are referred and you wanna skip the short queue you can go the private route but you'll have to pay fees which again you'll possibly be able to claim back through your healthcare provider or it may be covered depending on your policy and limits.
Unlikely your policy will cover dental. To get that type of cover on the healthcare you'd have to have the highest cost policy they offer and then add it on top as an optional extra. I get Southern Cross Regular Care with my job which covers all members of my family. To get dental it would cost me $120 per month by upgrading from the policy I have through work and then adding the dental cover. Too costly for me so we visit the dentist and pay. $400 for a filling. $100 for a check up.