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When you flip a light switch, the lights turn on almost instantly. Yet the electrons only move at 10e-4 m/s. A wire is not an empty pipe with no electrons; it contains many electrons. The electrons that flow through the switch the moment you turn it on are not the electrons flowing through the lamp immediately. What happens is that the closing of the switch causes an electric field in the wire, which moves very quickly and affects the electrons down the wire, causing current flow.
It is similar to if I had a long wooden rod and a button on the other side of the room. I can move the rod slowly, but motion travels through the rod at the speed of sound. So even though the rod may only be moving at 0.1 m/s, I can affect the button on the other side of the room in fractions of a second.
