I recreated Veritasium's setup in my simulator and measured the current through the load resistor, the results of which are here: https://imgur.com/a/sxVihf0

The gap between the wires is about 1 micrometer, so light should take about 3 fs to propagate through. The simulation output approximately matches this prediction, and over the first few tens of femtoseconds the current increases, with a jump at around 70 fs due to the reflected wave. All of this is pretty much in line with the results of Veritasium's experiment.

Thanks for bringing it up. I might include this as another example in my sim.