First I didn't go to school and have never been to said uni.
I think your options are snowplanet(ski dome outside of auckland) and Mt Ruapehu. It's like a 4 hour drive from Auckland though but there are always people going back and forth every weekend so carpooling would be easy.
Skiing is hit or miss. Epic in the spring but some pretty rough weather in the winter. Like ice coming down sideways and insane wind with no trees to block it.
At one point the upper mountain was closed for like 5 days. The weather reporting there is pretty off too. You really don't know what's going to happen until it does. You could make a plan to come down for the next weekend and then it could be shitty, 0 visibility raining and the park could be closed.
I was skiing ruapehu all season and I still had a good time. Longer season than the south island. Season is generally mid to late june- late october/early november.
A lot of people from england down there. I think you can kick it there for a while and they're chill because you're from england. A lot of english kids I met there were doing a 2 year trip.
and some other peoples would probably know.