The immediate issue is distribution at street level.
A few Teslas on a street, each charging at say 20kW (which itself needs an upgraded feed to the house, probably 3 phase) and the transformer at the end of the street will blow up
So there is no way EV will work for serious commuters.
20kW charging is really unusual for domestic EV charging. My PHEV has a 16A charger, i.e. 3.6kW. A bigger pure-EV car might have a 7.2kW onboard charger; some cars have 16A 3ph chargers which support up to 11kW but as you state would need 3ph. Most EV users will be perfectly cared for by a 7.2kW supply and indeed many will be fine on a 3.6kW charger.
The average UK commute is about 20 miles, assume both ways have to be covered by EV charging then 40 miles of range will require around 10kWh of energy to provide (4 mi/kWh). That can be done in under an hour and a half on a 7kW supply and within 3 hours on a 3.6kW one.
I think it's likely we'll need spot upgrades of the DNO network in places as the assumption is generally 6kW average per home, but likely only on crowded older estates.
New UK homes are also to be built with 3 ph supply directly which should help. Old UK homes generally have 80A or 100A 1ph supply; some homes are stuck on 60/63A feeds which will likely need upgrades as 32A car chargers are installed.
EVs will work for little cars doing short journeys, and a lot of people do that. What we don't know is how many people are willing to limit themselves to a little runabout vehicle, with little or no heating or aircon, when everything suggests people want the opposite. They could also run a big car like a Tesla but again only if they don't drive it much.
Well I do try to maximise the EV-only range on my PHEV but still use heating and air con. As EVs become more commonplace they will start to look and drive more like normal cars. Cars like the e-208/Corsa-e and e-Golf show that manufacturers are still interested in making "normal cars that happen to be electric". Which really is what we need, not weirdmobiles like the Leaf and Peugeot iOn.
And as Tesla has shown, there's a large market for luxury/high-end vehicles that are electric.