First, NZ is very food allergy and celiac aware - more so than the US. You often see signs in restaurants asking customers to inform the wait staff of food allergies, etc. There are also lots of gluten-free products available and gluten-free meals aren't that difficult to find in restaurants.
I'm allergic to peanuts. I find it much harder to buy processed food in the grocery stores in NZ, particularly ice-cream, crackers and cookies. They invariably have the 'may contain peanuts...blah blah blah'. I'm guessing those with treenut, milk, egg allergies (etc.) will have the same difficulty.
Also, lots of pre-made satay in the butcher shops and restaurant meals with peanut sauces, including pizza.
In the US, companies such as Kraft are so huge, they can afford to have dedicated lines that are peanut-free but not so in NZ with smaller manufacturers. Even Australian products are usually 'may contain'. When I'm in NZ I get a book-a-bach so that I can do my own cooking which makes eating much easier.
If you are peanut allergic - bring your own chocolate to NZ. It's difficult to find peanut-free chocolate.