I have done two tours in New Zealand since 2010, both on the South Island, so I can only make general comments about the North Island.
The North Island is hillier, more densely populated, the roads busier, and reputedly, the drivers more aggressive. I've heard negative comments from a few cycle tourists, but I've experience few issues myself. This may be because my visits have been in the spring when the roads are less busy. But generally the roads of the South Island have been mostly quiet, with the exception of State Highway 1, which runs the length of the east coast and is the main north-south artery. Plan to avoid this route as much as possible.
Your January visit coincides with the summer vacations, and is the height of the backpacker and tourist season as well, so it will be busier. It would be better to defer your vist by a month or even two if possible. The locals have gone back to school and work by February and the tourist season is starting to wind down by March, which is also likely to produce the most stable weather conditions.
Also, don't be deceived by the miniscule size of NZ on the map, depending on how far/fast you want ride and how much time sightseeing, it will take around 5 weeks to cover the South Island and maybe most of the North in the remaining available time. So far I have done two month-long tours to cover the middle and the bottom of the South Island, although to be fair I did go back over some parts that I enjoyed so much I wanted to do them again.
Last tour I rode several hundred kilometres on unsealed back roads. I found these roads well-formed and manageable (with just a little more than normal care) riding on 32mm Schwalbe Marathon Supremes, but the main touring routes are sealed. 1.6"x26" Supremes would be fine for your bikes.
NZ has a well-developed tourist route and it's quite possible to tour without camping if preferred, although you may need to do the odd 100km+ day. There are many motor camps (rv parks), backpacker hostels and camping grounds, and most towns have a domain where free camping is allowed, although there may not be much in the way of facilities.
The best touring guide is Nigel Rushtons Pedalers Paradise - there is a volume for each island. It's very basic but covers all the information you'll need with little weight penalty. There are very few roads in NZ so you don't need much in the way of maps or navigation - a simple road map from a newsagent or gas station is generally sufficient. I used Google Maps with an earbud to get voice navigation prompts passing through some of the bigger cities, which ofter have motorway approaches and convoluted cycle routes to avoid them.
Have a look through my touring journals - you'll find links and reference to the information I found most useful. Any questions you are welcome to ask.Chasing the Long White Cloud Gone Fishin'