Even if you include TOTAL energy instead of electrical energy (in the US, electricity is responsible for 40% of energy use, so I'll use that as a guesstimate), and assume 100MW/km^2 instead of 150 (even though I already accounted for that in the cloudy winter day part), you're still producing about 3.7 times as much energy on that cloudy winter day than the total UK energy use. And that's before we use higher efficiency solar panels or deep sea hydrogen seasonal storage (the latter of which is one of the very few sensible seasonal energy storage methods).

It's an absurd claim to say it's impossible, and there's almost nothing more annoying than a source claiming to be "without the hot air" actually being full of plenty of hot air on absolutist claims like this. You don't correct BS by using equal and opposite BS.

(Most energy advocates I know do this... both renewable advocates and nuclear advocates, exchanging BS about the opposite technology. It's a clean energy circular firing squad, with the climate in the middle.)