The speed at which electrons travel in a solid or liquid is 1 mm/sec. This is called drift velocity. This gives rise to the current in the wires or circuit. This will tell us how much current is flowing in the circuit.
When we turn a switch on, we are intantly creating a big potential difference (220V or 110V) across a bulb. The voltage difference is applied instantaneously all along the wire length , bulbs and any electric equipment connected in circuit.
So as soon as the switch is ON, electrons at every point on the wire in the circuit start instantaneously to travel at the speed of 1mm/sec.