Current at a point of the wire = electrons moving at the point. Electrons are present everywhere in the wire and push each other around. The electricity is detected at the cathode when the local electrons start moving, not when the electrons from the anode have come all the way there.

As an analogy: if you push a bar, the atoms in the bar will push each other around, and almost instantly, the other end of the bar is also in motion (the state of motion propagates at the speed of sound in the bar).

In the same way, push the electrons at the anode -> they will start pushing each other into motion until the electrons at the cathode are also moving. So the relevant quantity is the speed of sound in the electron "gas", which in most good conductors is close to the speed of light (in some fields of physics we can consider the collective behavior of the free electrons in the metal as a gas of sorts).