All velocities are relative to a specific frame.
You can describe the wheels motion relative to the vehicle body as purely rotation and no parallel translation.
You can alternatively describe the motion of the vehicle to the ground (assuming a basic cart going straight forward) as purely translation along a forward facing axis of the vehicle without rotation.
If you want the velocity of the wheel with respect to the ground. The vehicle is moving forward at velocity $v$ then the top of the wheel is moving at $2*v$ and the bottom of the wheel is moving at zero velocity then the axis is at the contact point on the ground. (The place where it has zero velocity.) And the rotational velocity is $v/r$ where $r$ is the radius of the wheel. And there's no translational component.
One critic part of the above theorem is that the axis of choice of the representation is not tied to the physical mechanisms causing the motion. In fact it may diverge significantly and also change moment to moment for a more complex trajectory.