The problem is that charging parked cars during the day will coincide with peak industrial demand and supply is already so tight during winter that they have had to pay major industrial users to drop off to avoid rolling blackouts. Prevarication over new nuclear build hasn't helped.
Charging overnight would work but then the generating capacity would have to run flat out on a continuous basis.
The distribution isn't quite that bad. It can cope with the likes of 3kW kettle loads on almost simultaneously in 80% of households when there is a major football final on at half time or a wimbledon final ends. But they have to prepare for it.
Compared to some continental electrical supplies where a 3kW kettle would fry the house electrics UK mains is quite robust.
This is correct in parts. Neutral is the balanced return loop and is usually somewhere close to 0v but not guaranteed to stay that way. Live is 240v on any one of three phases and local earth bonding is done at each premises. UK soil is wet so getting a decent earth isn't that hard. A 1m 1cm diameter copper rod driven into the ground will usually suffice.
A UK mains plug is polarised and has Live, Neutral and Earth - and there is a fuse in the live feed. Neutral and Earth are at about the same voltage most of the time but physically separate. Leakage current to Earth will trip the main circuit breaker in premises that have one.
Very old electrical installations have fuse wire only.
The big problem is that there is nothing like enough electricity generating capacity to provide all the extra power needed.
Street parking will be the killer for electric vehicles in the UK. Imagine how bad it would be with trailing cables running across the pavement (footpath) every car length in a country where people do still routinely walk between nearby locations. This is a daytime google street view - you have to imagine it with a solid wall of cars parked on either side of the street and with no preference as to where you park.
You don't get to park in front of your own house in street parking it is first come first served. Leads could be trailing for up to 30m.
There are chargers for Teslas and some other brands on motorway service stations but they are quite coy about their capabilities. I am actively considering an electric car now that diesel is out of fashion. Figuring out if I can cover the distances I need sensibly is a part of that.