At present it looks like Boris, Farage and Gove are standing around like deer in headlight, and the Labor folks are as usual confused running around lik a chicken with its head cut off. I don't think the Exiters expected Cameron to drop the entire mess in their laps and depart. I think they expected Cameron to invoke Article 50 on Friday and then be able to blame the entire mess on him. Cameron basically caught them flat footed since it is all their baby now, effectively. Surprisingly neither the Exit supporters nor the government had done any contingency planning covering this contingency.
But at the end of the day the effect on trains service is minimal to non-existent except that a vacation in Europe for the Brits is going to cost a tenth or so more, which might reduce traffic towards Europe a little. OTOH conversely for folks on the continent to visit Europe will be cheaper, but then who wants to go to England for a vacation anyway?