Some good info there from konev.
However, just to clarify, the Wakatipu Aero Club still exists even though it was kicked out of its home.
The commercial arm, Air Wakatipu, is operating from different premises.
For reasons that are too involved to recount here -- and which you might find hard to believe -- there is no flight training happening from NZQN just at the moment.
However basic and advanced mountain flight training can still be organised through the aero club.
In particular those North Island training organisations that used to send pilots to Queenstown for mountain flying training should not jump to the conclusion that it's all over now.
The club can put you in touch with instructors who are legally current and also still involved in commercial ops in the region.