In the Transformation constraint on the Tyre object change the Mix entry from Add to Replace.

Edit: Oops, just noticed that messes up the rotation around the z-axis.

In the Transformation constraint on the Tyre object change the Mix entry from Add to Replace.

Edit: Oops, just noticed that messes up the rotation around the z-axis. But you can use a Driver instead.

First remove or turn off the Transformation constraint.

In the 3D Viewport select the wheel(s) object. Then click the Item tab of the the tool-shelf panel (or N-panel). Right click the relevant rotation axis (x in this case) and choose Add Driver.

In the Driver window click on the driver (X Euler Rotation in this case). Then click on the Drivers tab of the N-pannel. Set the type of the driver to Scripted Expression. As the Expression write -var * 1.72787". As Driver Variable Type choose Transform Channel. As the object choose the WheelController (or MainController) Object. As Type choose X Location. As Space choose Local Space.

Now the wheels should rotate according to tot the orientation an direction of their controller object (while not rotating around a unwanted axis when the controller object does so; sorry for my initial advice)