OK if the motor is only two wire, then that takes "multi speed" motor out of the equation. Not sure why the switch would need more than to, there must be a speed control resistor somewhere. Sorry, but without seeing and playing with one, I'm just guessing.
Maybe this will help?
Help me rewire my sons Power wheels truck. - Ars Technica OpenForum
and.....
Modified Power Wheels - The Definitive "Wiring diagram" thread