My understanding is that although the actual electrons move relatively slowly, the effective change is almost instantaneous. The analogy my prof used decades ago was that if you lined up hundreds of ping pong balls in a trough, with no space between them, and pushed the one at the near end, the one at the far end would be pushed off the end almost immediately, even though none of them had moved particularly quickly.
Now, could you ever line up enough balls in a trough long enough that the the long distance divided by the short interval would be greater than 3 x 10^8 m/s? I think not. I think the speed of light is a real limit.
I'm not a physics guy - someone please correct me if I'm wrong, but my understanding is that the one force/entity considered instantaneous is gravity.
As well, Robert A Heinlein's classis SF story Time For The Stars treated telepathy as instantaneous.