Nothing moves the wheels. The wheels are simply freespinning. Yes, when the treadmill is moving backwards there's litte friection force applied to the wheels but way not enough to overcome the force of thrust delivered by the engines.

The force of thrus is completely not dependent on the speed of ground. No matter how fast the ground moves underneath, or whether it moves fast or slower, it would play no effect.

Let's say the same plane is on the same runway, BUT the direction of the runway movement would now be the same as the direction of the plane's take off.

Let's say that when the plane guns the jet engine the runway treadmill will spin at 400 mph ground speed (example). Let's assume that it would reach that speed instanteniously. Assume that there's no friction or rolling resistance.

The plane now is standing still and not moving anywhere. Engines are off. Now turn on the engines. The plane begins to take off.