OK, I'll have to look at a map. But off the top of my head, here are some things to consider:
Distances in the Sth Island are big. I'd choose a couple of places to concentrate on, and try not to do too much driving with a 3yr old. Flights are cheap, and consider the option of chartering a small plane - I once got a small plane from Wgtn to Awaroa Lodge for less than £100 return.
Personally I think Chch is not that interesting (for tourists, mind!) and would get out as soon as possible. There is a great beach nearby, can't remember name offhand, but very wild. I also like Akaroa, not very spectacular but lovely atmosphere - was a French settlement on the Banks Peninsula, has great restaurants. But if you want spectacular scenery, head for the Alps - Arthurs Pass is close, and not as touristed as Mt Cook or Queenstown, but you could justify going to all 3. You could even stay in an alpine hut at Cook and gawp at the climbers (very cheap!) There is an eco-lodge at Arthurs Pass which offers gourmet food and wildlife walks, expensive but very "authentic". Go walking on the glaciers, take a helicopter ride - your 3yr old will love it. Round Canterbury there are also some old gold-mining towns and relics, can't quite recall where, but interesting esp for kids.
If you don't mind the drive, I would go over to the West Coast and have a look at the forests there (another eco-lodge on this side near Haast Pass). Punakaiki is a fantastic rock formation, and you can go canoeing near there up the river into the bush, there are also good limestone formations here. Plus you can look at the Alps from the other side!
Wine - there are some good places to go around Canterbury/Queenstown. But the best place for wine is quite a bit further north, around Blenheim (about 4-5hr drive) - many of the vineyards have good restaurants attached as well, which by UK standards are very reasonably priced. Not quite so good round Nelson way, but the area itself is probably better for kids. The Abel Tasman National Park is a coastal national park with fantastic beaches, you can take a boat into Awaroa Lodge, stay there for a few days and go for day walks - a good trick is to take a boat down the coast and walk back. If you fancy sea kayaking, this is the place. If you keep driving round the coast, and over Takaka hill, you will come to a really lovely area with relatively quiet beaches. You can go to the "tip" and take a tour, where you will see lots of birds and seals. There is a lovely B&B near Collingwood.
Too much here for 12 days, but ask away about anything that particularly takes your interest. One thing to bear in mind is that this is peak holiday time for NZ'ers, so you may need to book accommodation esp near beaches (Nelson will be busy I think). Camping is a good option if you don't mind very basic facilities.