"That means if we switch on an electric bulb it should turn on after 17 mins. But we can turn on the electric bulb in our home at a lightning speed with a flick of a switch. This is because the speed of the electric current does not depend on the drift velocity of the electron.

Electric current moves with a speed of light. It is not established with the drift velocity of the electrons in the material. Thus, it may vary in material but the speed of electric current always established on the speed of light."

https://www.elprocus.com/drift-velocity-of-electrons-with-derivation/

Interesting read.

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Current at a point of the wire = electrons moving at the point. Electrons are present everywhere in the wire and push each other around. The electricity is detected at the cathode when the local electrons start moving, not when the electrons from the anode have come all the way there.

As an analogy: if you push a bar, the atoms in the bar will push each other around, and almost instantly, the other end of the bar is also in motion (the state of motion propagates at the speed of sound in the bar).

In the same way, push the electrons at the anode -> they will start pushing each other into motion until the electrons at the cathode are also moving. So the relevant quantity is the speed of sound in the electron "gas", which in most good conductors is close to the speed of light (in some fields of physics we can consider the collective behavior of the free electrons in the metal as a gas of sorts).

The answers here are mostly correct, however it's important to note that the speed of light is only c in a vacuum. In a typical coaxial cable, for example, the speed of electromagnetic propagation is roughly 0.75c. Still incredibly fast, but not the same.

With a pulse generator, an oscilloscope, and a suitably long coaxial cable, you can setup a demonstration and see this effect. A single short pulse will reflect at the unterminated end of the cable, and by measuring the time difference between the emitted and reflected pulses you can determine the speed of electromagnetic propagation in the cable.