The electron drift speed is estimated to be only a few mm `s^(-1)` for currents in the range of a few amperes. How then is current established almost the instant a circuit is closed ?

Explanation

Electric field is established throughout the circuit almost instantly (with the speed of light) causing at every point a local electron drift. Establishment of a current does not have to wait for electrons from one end of the conductor travelling to the other end. However, it does take a little while for the current to reach its steady value.