I am a kiwi in the UK, I am not in IT but I can give you some general advice. Now I grew up in the lower South Island and haven't lived there for 11 years so this may not be entirely right but the general gist is...
Start school at the age of 5, generally around your 5th birthday which means there are in takes all throughout the year, your dc most likely go into year 0 but may go straight into year 1 if born at the start of the academic year, i.e between Jan-March (our academic years runs from Jan-Dec with 3 sets of two week holidays throughout the year, followed by a 6 week summer holiday). Finish Primary at the end of year 6, do Intermediate school in year 7&8 and start High School in year 9 and start our High School Qualifications (NCEA) in year 11 until year 13. Private schools do intakes at year 7&9.
House prices in Auckland are ridiculously expensive, they are rising faster than London. I would try and move somewhere else if you could but my personal opinion is that AKL is not the greatest place to live, although I didn't grow up there so my view may be skewed. Sure there are some nice suburbs but they uber expensive and crime is super high in Auckland as well. Although AKL has lovely weather (be prepared for a lot of rain and humidity though), easy access to the UK, very multicultural and nice beaches.
RE childcare, ballpark figure would be $60 per day (may be higher in AKL), though once you DC turns 3 you will get 20 hours free.
Some general info: Healthcare isn't completely free, be prepared having to pay for GP fees and out of hours doctors. Not many houses come with central heating, food is more expensive - generally a more expensive standard of living
OP, just a couple of thoughts - living in NZ can be quite isolating, it almost killed me growing up which is part of the reason I now live in the UK. Say goodbye to cheap European breaks, its 2 hours to Melbourne alone. And in my personal opinion the NZ current Gov't is utter shit, it truly is rich out of the rich and there is not a lot of welfare support (there isn't even a child benefit!) and they chuck money away like no tomorrow yes I am looking at $26 million of taxpayer money spent on a flag referendum