The other car does 6 miles 3 times a week and is in the driveway almost all the time.
Sure, there are whales that do 300 miles a week, but that's not most people.
Looking at a sample of about 13 million vehicles that passed MOT tests in both 2018 and 2019, and thus reported the milage
Percentiles 20%: 2700 40%: 4800 50%: 6000 60%: 7000 80%: 10500 90%: 14000 95%: 17000
Lets assume that's all commuting over 5 days a week, 47 weeks a year. That means in an average 12 hour night:
20% of cars would need less than 11 miles of charge, or 4kWh -- 350W
Another 20% of cars would need less than 21 miles of charge, or 7kWh -- 600W
Another 20% of cars would need less than 10kWh -- 850W
Another 20% of cars would need less than 15kWh -- 1.25kW
Another 15% of cars would need 24kWh -- 2kW
All of that is less than a kettle, an average 13A plug is more than enough.
Now if those mileages include many people doing long distance drives at weekends and not regular commuting, the actual top-up needed every night isn't anywhere near as much. Some cars will do lots of miles at the weekend and need to draw the full 3kW out of a standard socket, sure. My kettle doesn't have an issue with doing that. For every car needing that full charge on a Sunday and during the rest of the week, there's a dozen that don't. OK, your battery might be down to 10% on Sunday night, but then you charge it to 20% for Monday morning, drop to 17% after your commute, charge to 27% Tuesday, and so on.
100A, or even 60A, house breakers aren't going to have an issue with keeping 95% of cars topped up, so you're at the next level. On the rare occasion you need to fully recharge really quickly, go to a specialist location with a 20kW+ charger. The data shows that most people won't need that most of the time.
Looking at newer cars -- ones made in 2016 (and thus having first MOT in 2019), you'd think they'd have a higher mileage. And they do, median is 8400, and 80%ile is 13,300, 95% 22700 -- about 30% higher in each bracket, but that's still well within a home charge range, even for the 95%ile, and only 1 in 4 cars are new enough to be in that category, so median mileage of all cars would be under 7,000 miles a year range.
You're right there's a grid level problem -- the dip in power generation in the UK overnight is about 10GW, which could sustain about 14 million cars - not too great when there's more than twice that needing power.
Assuming that in a given area cars are distributed fairly evenly as above, with some cars needing 24kWh a night, but others needing 4kWh, the average overnight draw would still only be 700W, that doesn't feel like it's going to trip neighbourhood substations, but further upstream could be problems.
