How can I replacing broken DPDT switch with direct wiring?
The blue and white wires seem to be the contacts of the motor and the black and red is the battery.

If you observe the way the black and red wires are connected, you can notice that they are one way in the top connectors and flipped in the lower ones - thus providing the reversing function.

If you connect the middle two prongs to the upper two(individually), the car will move one way (cyan in image). If you, however, connect the middle two prongs to the lower two(again individually), the car will move the other way (magenta in image). 
Naturally, connecting cyan AND magenta at the same time will result in a shorted battery, so don't do that. :)

You should get two wires and jump the connections as indicated in the image. Do make sure the wires make good contact or the contact surfaces might get hot.


Schematic for the footswitch:







simulate this circuit &ndash; Schematic created using CircuitLab

Shorting the motors out when the footswitch is inactive makes the car "apply brakes" (shorting motors out results in them resisting rotation).

Drawing out the schematic also explains the second blue wire in the mode switch.