Electrons don't need to bump into each other to transfer movement, like billiard balls. They are charged particles which interact with each other at a large distance through the electromagnetic field.
Imagine a room packed with balloons: if you push on balloons on one side of the room, the wave reaches the other end of the room in a matter of seconds, even if none of the balloons traveled more than a few centimeters. In this analogy, balloons represent electrons together with the field they produce.
This is why the electrical signals travel at e.g. 0.7c, while individual electrons move at less than 1% of that speed (depending on the temperature), even at microscopic level. At macroscopic level, the charge drift happens even slower, with typical speeds in mm/s or cm/s, depending on the charge and current densities.