I believe this is the same video, the author theorized (probably correctly) that it was because there was residual 'power' on the wire, as it acts like a giant capacitor, among other things. this is because the wire that immediately showed up was extremely small and dissipated very quickly. https://www.youtube.com/watch?v=2Vrhk5OjBP8
Someone set up some long wires over a farm field, and measured the time to first current flow, which was faster than if the current had to travel the whole wire length. Granted, the full current flow doesn't appear until later, but the first little bit of current is there, which is exactly what Derek said in the Veritasium video