Though the drift velocity in a conductor responsible for current is very small, the light in a room turns on immediately after the switch is turned on. Why and how does this happen?

The electron drift velocity is the average velocity that an electron, attains due to an electric field. Due to the collision between the electrons, the average velocity ie electron drift velocity is very small. when the switch is on, an electric field is established at a speed of light, it exerts an electric force on every free electron in the conductor that makes them all move with a large acceleration. In other words, the electric field takes place with the speed of light and the free electrons everywhere in the wire begin drifting almost at once. So, though the electron drift velocity is very small but electric bulb lights instantly when switched on.