The car doesn't necessarily use the speed and front wheel direction to estimate the position. You can get portable GPS units that can do estimates just like the Tesla does, even some phones. If you don't use the speed and front wheel direction from the car, you need at least one accelerometer, and one gyroscope. Using the starting speed from the GPS, the accelerometer is used to determine the change in speed, as deceleration/acceleration is measured. And the gyroscope is used to determine the yaw of the car, giving you the direction of travel.
More commonly, a three axis accelerometer as well as a three axis gyroscope is used. The three axis gyroscope not only gives you yaw, but also pitch and roll, and the three axis accelerometer not only gives you longitudinal acceleration but also lateral and vertical acceleration. The orientation of the device also becomes irrelevant, as you can calculate the actual movements of the car based on the gravity and the direction of travel obtained from the GPS before entering the tunnel.
The above system is called an Inertial Navigation System (INS) and is used particularly in aircraft and missiles to complement the GPS. The accuracy varies, but a good INS can be quite accurate up to around an hour of operation. (It gets quite complicated as time progresses, as you need extreme accuracy in the measurements, and you need to take into account things like the rotation of the earth.) Any cheap INS should handle a tunnel fairly well.
