There were 240.0 billion car miles driven in the UK in 2013. That’s 386.2 billion car kilometres. [source]
If electric motoring costs 0.34 kWh per kilometre (from your 4 year old link), then that requires (386.2 × 0.34 =) 131.3 TWh of extra electrical energy.
Total electricity generation was 359 TWh in 2013. [source]
So switching to all electric cars would required the UK to increase electricity generation by 37%, according to the article you linked. If we all drove Teslas, which achieve 0.24 kWh/km, then generation would only need to increase by 26%.