No, just electromagnetism. If a charge is moved, the strength and direction of the electric and magnetic fields in the surrounding space change as well, and the perturbation travels outward at the speed of light. So in Veritasium's circuit, the charges near the light bulb on the far end of the wire, just 1 meter away, feel a force due to the movement of the charges across the closed switch, and they feel that force just 1 meter divided by the speed of light (about 3.3 nanoseconds) later.
This initial movement of charge does not represent the full current (whatever the battery voltage is divided by the total resistance of the circuit), which takes longer to build up, since the electrons all throughout the circuit take some time to start moving and reach a steady flow.