In reply to Eric the Red:
Hi. I'll start you off...
Arthur's Pass is mid south island and a popular spot for "less serious" mountaineering, which in winter could be likened a bit to Scotland. You can do small peaks, or ridges with just ice axe and crampons, but you need to be pretty good with a map and compass to do anything in NZ really. Especially in winter.
AP includes Mt Rolleston, which in winter is loads of fun. You could get yourself up (the low peak anyway) with excellent snow/ice/navigation techniques and weather forecasts. But Mt Bealy, or Avalanche Peak are good snow/ice walks that will give you good views and a feel for your abilities and the area.
http://www.softrock.co.nz/mg/index.php
There are loads of multiday walks in the area too. Which will get you a million miles from anyone in about 10 minutes.
You often need to carry snow chains or have a 4WD to get into places like this in July. It's not deep snow that's the problem - unless you are unlucky. Just ice. No salt grit in NZ.
Camping or hutting-it is fairly cool, so yes, a good bag is pretty essential. But it is not antarctica. Nor is it quite as cold as Scotland in winter.
Unfortuantely, I don't know about the "Great Walks". Reading this about the milford track:
http://www.doc.govt.nz/parks-and-recreation/tracks-and-walks/southland/te-a...
I wouldn't recommend it to someone who hasn't spent quite a bit of time on easier winter stuff in NZ first. Tramping in NZ in winter requires pretty good survival skills. You don't get that sort of experience in the UK.
Have fun. Be safe.