Imagine that you have a hose already full of water and connected to a tap at 10m. The water pushing behind the tap causes water to flow out of the hose’s end immediately after you open it. Although the water molecules from the tap might take some time to reach the lawn sprinklers, your lawn will start getting watered once you open the tap.
The same thing happens with electricity: a wire already has electrons. It doesn’t have to be filled with electrons by connecting it to a source (flipping a light switch). It just pushes electrons into the wire. These then push the electrons in wire forward (the “push”) travels at the speed light in conducting wires, not the electrons. The electrons *already* in the wire are what heat the bulb.