An electric current is a flow of electric charge around a circuit. The charge is already in the wires (carried by billions of tiny particles called electrons). This charge is evenly spread out through the wires. As soon as you close the switch, the cell starts to push on the charge. So all the charge starts moving at once.
It's a bit like a bicycle chain. The links are like the charge, the wheel is like the lamp and your feet are like the cell. As soon as you start pedalling, the back wheel starts to move. This is because turning the pedals makes all the links move at once. It's not just the links nearest your feet that move.