The drift velocity of electrons in a conducting wire is of the order of 1mm/s, yet the bulb glows very quickly after the switch is put on because 

The random speed of electrons is very high, of the order of ${10}^{6}m/s$

The electrons transfer their energy very quickly through collisions

Electric field is set up in the wire very quickly, producing a current through each cross section, almost intantaneously

All of above