Surely the whole point of having a "national" grid, is that there is no single "generating component"? We already have multiple generating components - thermal power stations, nuclear power stations, wind farms, hydro, etc.
As for the installed cabling, it's obviously sized to be able to cope with peak demand at present, so for all the time EVs are charging off-peak, it will never be a problem. Locally, I can believe that some substations that are already running close to capacity, might have to be upgraded, along with their associated cabling, though.
Of course, that's where smart meters and V2G come into their own. I would expect the distribution companies to request short term switch-offs of some charging EVs (via the smart meters), in areas where demand was getting excessive. All in exchange for money, of course! What I envisage, would be a situation where the distribution network operator sends out requests to a number of smart meters, which will then notify their occupants of a request to stop charging - probably with appropriate sweeteners like a few kWh of free electricity, some hours further ahead. The house occupant would then be able to accept or reject the offer, depending on when they felt they needed the car.