I have finally added the differential effect to the wheels. At first I tried to use the rotation of the main drive control bones as a factor for my rotation driver, but that only worked going into the turn, coming out it would revers the desired effect!
In the end I placed two new bones perpendicular to the chassis bone (for the back wheels) and the front wheel direction bone (for the front wheels). The z rotation of those bone does work. Now it will react to all the turns automatically along the drive curve. Here is a new video (I have not updated the old ones)
(edit I changed the video)