Both me and DW had pain in the wrist and top of the thumb after DS1. He was very heavy and picking him up caused a sort of RSI or carpal tunnel syndrome.
I suggest you get an elasticated wrist support bandage from a chemist or a sports shop. It will need to be fairly tight BUT do not wear it 24 hours a day takeit off for a while.
Sounds like you have the same conditon as me and DW had. Gripping the steering wheel will have been enough to exacerbate it. Your GP was wrong.