Electrical Power production response to complete transition to electric vehicles
It doesn't matter.

Burning fuel to charge batteries as the absolute worst case has somewhere between similar, and slightly better efficiency than burning fuel in a car, as the motor can always be run at optimal efficiency.

For trains, this has been done for ages -- quite a lot of diesel engines are really "serial hybrid" setups where a diesel generator feeds an electric engine, as this is more efficient. This is also being done for some cars targeted at customers that require individual mobility with long range.

This is the worst case, however. Since electric cars do not require grid power while they are running, but rather when they are idle, it is possible to schedule the charging cycles during off-peak times. Electricity companies are already introducing remotely managed charging ports that are turned on only for a few hours at night, and house battery solutions that can be charged from a rooftop solar panel, all of which reduce the strain on the grid that is introduced by electric cars.

Total demand however still increases. Right now, excess energy produced at night is stored by pumping water uphill, when we charge cars instead, this energy will be missing throughout the day, when consumption exceeds production.

There are also be non-technical effects on grid load, such as telecommuting, online shopping, automation in manufacturing (there are several factories that usually run in the dark as there are no humans inside), ... All of these effects combined shape how and if the grid changes to accommodate demand.

I'd expect that in the next ten years, demand for individual mobility will go down quite a bit, while logistics demand will rise even further, so many electric cars will be commercial, and they will be charged over night in industrial areas.