One problem is that our local electricity supply grid is sized and rated for a typical household load, although it varies a bit with the age of the local infrastructure. It was common to find the per house demand estimate being as low as 8 kW around 50 or 60 years ago, for example, and although I believe that the current standard for demand planning is a per household demand of ~15kW, in practice there is diversity applied to the LV grid, so the DNO never expect that all houses on any particular phase will draw ~15 kW at the same time.
It's worth noting that the fuse in the company head is usually rated at 60 A, 80 A or 100 A (ignore the fuse holder rating printed on the case, that pretty much always reads 100 A). The fuse is very much a last ditch protection measure for the cabling, not the rating of the supply. having the highest available single phase fuse fitted of 100 A doesn't mean you can draw 100 A from the supply all the time, it just means that the cables are rated such that they will be protected if 100 A is drawn from the supply for short periods. Our supply (put in 7 years ago) is has an 80 A fuse, for example, which isn't untypical. We could have this uprated to 100 A, as the cables are sized for that, but in reality 80 A is more than enough.
In general, household demand has dropped a lot over the past ten years or so, due largely to changes in the stuff we use. Appliances now tend to use far less power than they used to, with everything from washing machines to TVs having been made a lot more efficient. It's also worth looking at how much energy we really need to run an EV. Someone doing a long commute each day (as I used to) of about 50 miles each way, would need to put around 30 kWh back into their car every night, assuming a fairly generous 300 Wh/mile. Assuming they had at least 8 hours charge time overnight would mean that charging at just 3.75 kW would be fine, around half the maximum charge rate we can get from a typical 32 A charge point.
Alternatively, we could look across the whole country, and try and estimate the impact of most cars being EVs. The average mileage driven in the UK is about 10,000 miles a year. If we assume that this is only on weekdays (makes for a worst case example) then that is about 38 miles per day, or about 11.5 kWh. That's barely enough to justify having a 7 kW charge point, really. The effect of drawing another 11.5 kWh per day per household in the UK isn't that great, especially given that domestic electricity demand has dropped a lot.
There are currently about 31 million electricity consumers registered in the UK, up from about 27 million consumers in 2000. Despite that increase, demand per consumer has reduced a fair bit, from ~12.2 MWh/consumer/year in 2000 to 9.52 MWh/consumer/year in 2019, a reduction of around 22% per consumer.
The ~2.68 MWh per consumer that has been saved, due largely to changes in the energy efficiency of appliances, I suspect, is equivalent to about 8,900 miles of EV motoring per consumer, at a fairly generous 300 Wh/mile. I suspect that, in reality, this energy saving alone may well be close to enough to cover the real energy cost of running one EV per consumer, given that many EVs seem to be able to get better than 250 Wh/mile.
Generation capacity is also increasing, as more and more renewable generation sources come on line, and the grid has got much the same diurnal balancing problem that it has had for decades, in that day time peak consumption is massively higher than the night time trough. Decades ago we came up with the idea of night storage heaters and various off peak tariffs to try and redress this balance problem, but now it looks very much as if charging EVs overnight might help with a part of it. Charging an EV overnight, every night, isn't as big a demand on the grid as charging storage heaters used to be (they would typically charge at >10 kW per household for 7 hours every winter night) but it will help.
Hope the above has helped put the energy supply situation into some sort of perspective, as I often hear arguments from the anti-EV brigade that all this "extra" electricity has to come from somewhere, so will need loads more power stations. The reality is that the true impact will be pretty small, as EVs just don't make that much difference, really.