10-11 Sizewell C's, assuming the UK 2025 average demand of ~33 GW.

A couple more to deal with outages. Plus some battery capacity, which you want because nuclear doesn't load follow well. At £38 billion per, you need about £500 billion to power the whole country. About 40% of one annual UK budget.

It's possible to imagine building Sizewell C's at considerably lower cost, as is done it other nations that exist, and manage to build such things without the huge multipliers, but that's another matter.

There are nuclear designs that follow load just fine. The UK just didn't build them. Also, the designs that load follow are usually liquid fueled reactors, not solid fueled ones which are normally built. PWRs generally have the lowest utility costs (the actual cost you pay, not the cost to build which is normally quoted) of any power source we have tried. Even the fast-sodium reactors the Soviets built can't match their price. That's part of why building stock nuclear designs works so well (eg the French) but trying to build experimental reactors doesn't. And when you try a new design, you have to build a pilot plant first to work out the kinks. The British always skip this step and this shows in the results.