Electric batteries don't power anything, they're best compared to empty buckets.
In analysing the effects of increased usage of electric transport, what's important is where new electricity generating capacity will come from. In the US that's primarily natural gas and solar. Coal generation has actually been decreasing.
Of course, that's not to say that if everyone suddenly started driving electric cars that all the new supply would be green energy.
However, even coal power stations taking into account powerline losses are more efficient than a typical car's engine, and in Western countries usually cleaner as far as particulate and nitrous oxide pollutants are concerned.
Coal's share of electricity production is declining or being phased out almost everywhere.
Coal still produces 40% of every KWh consumed with Nuclear making up a fair portion of the remainder.
Renewables are at 20% and (fortunately) rising sharply but for the moment fossil energy sources are by far the bulk of the energy consumed by electric vehicles.
Even in countries that still have coal-fired power today, an electric vehicle is certainly no worse than a fossil-fuel one.
And, as grid share of renewables and other cleaner energy sources increases, each EV becomes greener and greener over time. But each fossil vehicle remains just as polluting - or perhaps even gets worse as it ages.
I've yet to see an end-to-end over the lifespan of the vehicle report that was factual enough that it was possible to make that call.
Keep in mind that EVs have been subsidized a lot, that the batteries themselves are made with technology that also pollutes and emits CO2 and that a new EV is almost certainly a net negative compared to continuing to driver an older ICE because the ICE car already exists and the EV has to be built new.
That makes the whole equation so complex that I really do not feel comfortable saying that an electric vehicle is certainly better or equal to a fossil-fuel one, it would very much depend.
I'm writing this as a long time renewable energy nut that would very much like to see our dependence on oil reduced to manageable proportions and not for something like transport where we have other alternatives. Even so, if this is all to succeed then the best way forward is to be realistic rather than to hype EVs and the reality is that right now people claim their EVs are 'green' when in fact they are to a large extent turning decentralized pollution into centralized pollution.
Thermal efficiency for a coal plant is ~35% give or take (depending on how new the plant is) so that would be your starting point.
Charge/discharge losses for a large (20KWh) pack that I used were around 25%, a modern EV might do a bit better but probably not that much better (because the internal resistance of those cells they use is quite high).
Putting all that together:
.33 * .75 * .95 * .97 = ~22%, and ~24% for a high
efficiency coal plant.
So it's actually pretty close, better than I expected.
Right now new fission power is not economic because of enormous capital costs---but most of that is quite negotiable. Nuclear could be much cheaper, if we only held it to the same standards as eg gas. (I am not even proposing holding it to the same standards in terms of life lost per Joule as coal---coal is aweful on that metric.)
These things were all prohibitively expensive for most people in Western countries, but their declining cost has made them more affordable and more used.