Where does the Arduino fit in the car?
Seriously, there is not enough information to provide meaningful suggestions. We would need to have a datasheet for the motor to even start and it may be impossible to get that level of detail.
Servos are usually not 12v. 5v or 7.2v is more common. Servos usually run from 2S or 3S batteries. How was the servo powered before? How does breaking the back H bridge affect that supply?
The Arduino is going to fit where the old board used to be, the plan was to remove the chip and take control of the old board and work from there, although the h-bridge which controls the forward and backwards of the motor doesn't work.
The servo is a 9.6v servo I think.
You can purchase electronic speed controllers for cars from HobbyKing or some other supplier.
You want to make sure you get the right kind.
If you provide a link to the car you're using, we could probably help you select an appropriate ESC.
I second MoganS' question about purpose of the Arduino. If you just want a RC car, you don't need an Arduino.
Depending on the electronics of your car, you might need to replace the radio system.
I am using a Tyco Street Beast from around 2004ish.
This is the best information I can find for it.
The Arduino will be used to control the car entirely, and make it autonomous, therefore I'm likely to add things like GPS, or collision detection at a later date. Therefore I need to use as little of the Arduino pins as possible.
Also, the old board is completely toast. I had the car in storage from when I was kid, and it seems to no longer work since then, only the servo responds from the chip. I would like to replace it anyway, as I can always re-use the board for other projects.
Also, the Arduino is an Arduino Nano as I originally intended to place the Arduino under the original board, so Motor shields may be out of the question? I just need a H-Bridge really to drive the one main motor backwards and forwards. The servo already has a h-bridge built in I believe although haven't checked, I'm mainly assuming because servos I've repaired in the past have had them built in although they are newer.