There's a pretty large over capacity in the UK as shown by the daily difference between peak and overnight power usage. Here's the details including every kind of power produced, imported, stored, and exported:
http://www.gridwatch.templar.co.uk/
There's a 15-20 GW difference between daily peak and trough - that could charge many cars overnight.
If you look at that dashboard closely, the huge variance is provided by CCGT and they are not a source you want to be pushing 100% all the time. Also CCGT, in the UK, at peak demand, is nearly at 100% capacity. CCGT needs much more maintenance that older larger coal based or nuclear based generating stations.
Moving fossil fuelled vehicles to the grid would mean putting the grid up to 100% 24 hours a day. Not something the generating infrastructure is designed for.
If we were to move even 10% of the vehicles to EV we'd be pushing the grid past breaking point.
Also you have to factor in winter and summer too. As the summer peak heat increases people buy air conditioning. Pushing peak power up in the evenings in summer. Peak power in winter also increases due to electric heating.
There simply is not the capacity in the UK grid to support significant EV and if we tried we would start to have generation failures.
These things have to be planned and no planning has gone into the transition from oil based transport fuel to electric based transport fuel.
Over and above that, we're talking humans here. We only need 10% of them to charge during the day, after forgetting to charge at home at night, to bring the grid down. Chances that people will even realise what they are doing in mass adoption? Slim in my experience.
I would like to see EV take up in the UK. But without a concerted effort to provision the energy required, then I can't see it as feasible.