When drift velocityis so small, how is it that an electric bulb lights up as soon as we turn the switch on?

Explanation

A particular electrons does not have to reach the electric bult to make current flow in the bulb. As soon as the switch is turned on, the potential difference gets applied at the ends of the terminals of the bulb and an electric field is set up.
Changes occur at the initial state of setting of electric field which give rise to a changing magnetic field. the changing magnetic field gives rise to another changing electric field and so on. this electromagnetic phenomena takes place at the speed of electromagnetic waves which is the same as velocity of light `3xx10^(8)ms^(-1)`.
The set electric field make current flow through the bulb instantly.