If mum has a walking problem flying back will be very tiring as walking to and from gates and to parking either end is quite far unless you can borrow an airport wheelchair. PM if you run into trouble in Wellington.
I would suggest contacting the airline and arranging for assistance. Much easier to travel if you have some help. Should not be scared of asking IMHO.
Here is more information about Special assistance at Wellington Airport.
Would probably do Auckland to Wellington by train and Wellington to Auckland (or vice versa) by plane if I was doing the journey.