Rather than doing simplistic calculations claiming to show that vast new numbers of power stations will be required, why not look at National Grid's own forecasts?
https://www.nationalgrid.com/group/case ... future-evs
They point out that it is not at all sensible or realistic to assume, as T_N did above, that the large predicted numbers of EVs will all try to charge at once at peak times. That indeed would need an impractical number of power stations. What they are planning is smart meters that measure and charge in half hour intervals throughout the 24 hours, linked to the price and availability of power. They don't seem to be concerned about coping, as long as the 'smart grid' is brought in nationally, plus maybe a 10-15% growth in generation power,
It does not take all night to charge a BEV. The 40 KW Leaf takes about 5-6 hours for a full charge, and most days many drivers will only need a popup charge. It is already possible to use an app to control when this is done so as to get the cheapest power.
And this is not futuristic hopefulness. I have just ordered a smart meter from Octopus and a charger from EO that will do this already, and also take account of when solar power is available from my roof panels. The Octopus Agile tariff charges in half-hour intervals. Most of the day it is around 5 - 6p a unit, in the small hours it is 3 - 4p but between 4 and 7 pm it is about 24p. It doesn't need legislation or enforcement to persuade people to charge when the power is cheap. This compares to roughly 15p a unit charged by the big power utilities and even Economy 7 is 8p.
And in those peak hours, it is already possible in some areas to sell your surplus stored battery power back to the grid (Ovo and Octopus have extensive trials working now). The combined storage power of, say, a million BEVs that have 80% charge left at 4 pm is in the region of tens of GW - that's roughly five major power stations or Dinorwig hydro storage installations, which will make a huge contribution to the grid. And there is plenty of time after 7 pm to charge your car ready for the morning.
BEVs are not the complete answer nor the answer for everyone. Hydrogen or (perhaps better and safer) ammonia fuel cells may be the best for heavy transport, though they are still a long way off making a major contribution. They will not alone fix the CO2/greenhouse problem in time. They will make a serious contribution to reducing CO2 (yes even when you take manufacturing costs into account) and a massive contribution to cleaning up air pollution in cities. Of course many other changes will be needed, especially in agriculture and diet and in consumption generally.