We arrived on 20th September and started out from Christchurch on the of 23rd September. It was 18 degrees when we left and nearly 2 months later it is 14 degrees today. I guess what I am trying to say is the weather is very changeable from day to day or even hour by hour. Although the average temperature has increased slightly a sensible cyclist will still pack for cold and wet weather if leaving if starting their tour in November.
A good set of waterproofs, warm clothes and a decent sleeping bag will see you through the weather. The big bonus is that almost all of the campsites have camping kitchens giving you a place to warm up and relax. At this time of the year we found that the campsites were deserted and we often had everything to ourselves or one other group. If it is really pouring down hostels aren't too expensive and many campsites have cabins which vary from fully self contained luxury to garden sheds. They are however warm and dry.
At night I slept in thermals and a cheapish sleeping bag with a rating to -12 extreme -5 comfort but Maria had a good quality bag rated the same and didn't need thermals so draw your own conclusions. Packed my bag was only 2/3 the size of hers though.
Our rule was if it rains no camping, but after a few nights in cabins we got a little too comfortable and only broke out the tent when we had too. One of the reasons was the YHA offer a great discount to no supported cyclists $19 for a dorm $58 for a double where ever you stay, in Mt Cook a double should have been $110 so a massive saving. If you decide to go the hostel cabin route it may be best to make a reservation especially later in the season. It will save you carrying alot of gear and therefore make the climbs more fun.
Over the last week or so we have seen more and more cycle tourists where we only saw 2 or 3 in the first month. We also travelled anti clockwise where most of the guide books say clockwise due to the winds. I can honestly say we had at least as many good wind days as bad so either way round would have been fine.
In conclusion I would say the weather hasn't changed hugely in the 2 months we have been here, nights have got lighter but nobody has turned the heating on and the water off! With that in mind I guess it is what you are looking for. Quiet roads, accommodation and lower prices can be found in the early part of the season with the flip side being a more social trip if you decide to ride in the summer.
I wouldn't leave any earlier than mid September and many places had said we were the first cyclists they had seen that season but we were by no means they only pedal pushers out there.