Although the drift velocity of electrons is very small (only a few mm/s), the electric field produced by the battery is established throughout the conducting wire almost immediately after the circuit is closed.
This electric field propagates through the circuit nearly with the speed of light. As soon as the field reaches the free electrons present everywhere in the conductor, they start drifting simultaneously.
Thus, the current appears almost instantaneously throughout the circuit, even though an individual electron moves very slowly.
Therefore: