The rate of flow of electrons is known as current. When we switch on a light bulb, then it lights immediately. If drift velocity of electrons is so small--at least on the supply on 220V (supply of my home)--then why does the light bulb light up so quickly?
Think of the electrons as a line of soldiers ready to march.
At the commander's call, they all start moving. Each individual moves slowly (drift speed) but they all move simultaneously. So the motion is seen at the front of the line right away.
How quickly the front starts moving depends on how quickly the signal reaches it. That signal being the the commander's call (at the speed of sound) or the propagation of electromagnetic waves when the power source is turned on (at the speed of light).
In short: The electron delivering the power to the light bulb is not the one that just exited the power socket. But the one just exiting the socket is "pushing" on the next, which "pushes" on the next, which "pushes" on the next ... until the one in front is being "pushed" through the light bulb filament. This "push" as an electromagnetic signal propagating at the speed of light is what causes the reaction time - not the drift speed.
Because the electrons crossing the closing switch don't have to travel all the way to the bulb to make it light up - they just have to push other free electrons in the wire and get the current going in the whole loop, which extends to a transformer and, in a way, all the way back to the power source.
A similar phenomenon could be observed in a water heating system. The pipes are always filled with water (as wires are filled with free electrons), so as soon as the pump is turned on, the water starts flowing in the whole loop.
There is a relatively small delay in both cases that has to do with the finite speed of the propagation of electric or pressure waves, but that delay is very small in comparison with the time it would take an individual electron or water molecule to travel all the way around the loop.
The electrons are already within the material, they do not need to travel from your power socket into the light bulb. It is rather that all the electrons in the wiring and the bulb start to move roughly at the same time. So even if all these electrons move slowly, it is a collective motion everywhere in your circuit.
The conduction electrons that are already in the wire are moved by the electro-magnetic wave that can propagate once the circuit is closed.
With a 220V ac supply, actually the electrons are moving back and forwards, their net drift velocity is zero.
There are many other analogies to this. Sound moves in the air at 300m/s, but it is not an individual air molecule that travels from the source to your ear in that time, but a wave of pressure.