New Zealand - October or December?

It's been a few years since we were in New Zealand over the December holiday period but the only day I recall there being an issue with closures was Christmas Day itself - otherwise it was pretty much business as usual.
There's a lot to be said for traveling during the "shoulder season", which Sept/Oct would be...cheaper prices, easier availability, etc. That said, it will be much warmer in December and generally more pleasant to be outdoors, and I don't recall there being that much difference regarding rainfall totals.
NZ is about the same size as the state of California, so the further south you go in early spring the more likely you'll be to encounter lingering snow in the mountains if you're going to someplace like Queenstown. Some of the popular hikes may still be closed too...am thinking of the Routebourne Track in particular. Still, Milford sound, Te Anau, etc will still be open for business and will be much less crowded in early Spring. North Island is more tropical and won't be as affected by the seasonal change. Traffic, especially on the South Island, really is a non-issue once you get away from the Auckland and Wellington metro areas.
Suggest doing some comparison shopping by plugging in your proposed dates and then deciding what's best for you.

I would find out about weather. I was surprised at how mild the temps were in mid to late January, while "nearby" Australia was having a hotter-than-average summer.

as a comparison, parts of the south island are lower than tasmania, whereas auckland, in the north island, is about equivalent to sydney, so weather wise it really depends on where you are in NZ. late sept/oct could be nice temp wise, for travel, but too cold to go swimming etc. nov or early dec would be good option, if possible?

in australia there are not too many businesses close down ( other than industry) over the xmas break. NZ relies very heavily on tourism so they will probably open most days.

my concern would be the locals would also be on hols at xmas time.( eg: booking accommodation) whereas sept/oct 'might' only clash with local school holidays? you would have to check.

particularly in the south island in late sept , they would still be skiing, snow on the mountains and very beautiful. as a guide the ski season in OZ finishes in early Oct

I would investigate spending time around Queenstown , so much to see/do.

hope this helps.

We went in December a few years back as a family of 4. It was a great time. Expericed some chilliness and rain in South Island - nothing that stopped us from doing all our outdoor activities, but North was sunny and warm. Big city and some very touristy spots are crowded but a couple of years back at that season the "crowds" were nothing compared to what you see in Europe around that same time. We did not spend too much time in big cities as well. We rented a car and drove all over. Nothing was closed.