Question

how is it so that a bulb glows as soon as it is switched on; though the electrons drift at a speed of about 1 or 2 cm/s.

Answer 1

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 electric force on every free electrons in the conductor that makes them all move with a large acceleration. So, though the electron drift velocity is very small but an electric bulb lights instantly when switched on.