There's a very useful dashboard here that gives the carbon content of UK electricity at any given moment of the day. The figure given is the full 'source to socket' value, and includes emissions from mining, fuel transportation and grid losses. It then goes on to compare electric cars charged with that electricity with the 'well-to-wheel' emissions of a few petrol and diesel interal-combustion-engined (ICE) cars, which includes both tailpipe emissions and emssions from refinery, extraction, crude transportation and so on.
At 15:30 today, the carbon footprint of 1 KWh was 664g. At that rate, the emissions of a Nissan Leaf were 102g/km, compared to a Toyota Prius (the best ICE car, albeit a hybrid) at 107g/km. A Mitsubishi iMiEV (electric) gave 66g/km, but this is quite a small car. A Mini E (electric) gives 81g/km, whereas a Mini diesel emits125g/km.
On this basis, even in the middle of the day when electricity demand is high, electric cars would appear to give significantly lower CO2 emissions. There are a few other things to consider however.
1. The 'last megawatt problem. If you plug in an electric car, you are increasing the load on the grid. At some point additional capacity will be spun up to cope with the load, and that is likely to be the highest-emitting generation capacity. Therefore, correct accounting means you should assume that the electric car is essentially coal-fired, which increases the attributed emissions significantly.
2. Most drivers of ICE cars struggle to get anywhere near the manufacturers' claimed fuel consumption and therefore CO2 emissions. That's partly because of poor driving habits (sudden acceleration, excessive speed, poor anticipation), and partly because the headline CO2 figures are obtained by experts in a lab rather than in the real world.
3. Currently there is little information on how achievable are the manfacturers' claimed ranges for their electric cars. Like claimed tailpipe emissions, they are subject to all sorts of real-world factors, such as using the heater/demister/air conditioning.
4. The figures given for electric cars don't seem to take into account charging losses.
Once again, the environmental case for electric cars charged using current UK electricity is some way from being compelling, unless you ignore the 'last megawatt problem'. While you are unlikely to be making CO2 emissions worse by buying an electric car, it is by no means a passport to guilt-free motoring.