Have you ever asked yourself why this is? There are two basic types of reasons why the UK nuclear industry does so poorly. The first reason are scientifically illiterate "activists" that delay and block construction. But this is only part of it. The other and possibly bigger part is that the UK doesn't build stock nuclear plants. They "Wallace and Grommet" things far too often. In this case, that means not building proven designs but instead building experimental ones. Specifically building high temp reactors which haven't been built before. Normally (at least in the US) we build pilot plants (think 1/10th the normal size, so 100MW or so) and then once we get the kinks worked out, we built the full size plant. In the UK they just built full sized experimental reactors. This is why the UK has so many problems with (the cost of) their nuclear plants. Had they built a stock design (like the French) they wouldn't have these problems.

What made the cost to balloon so much, and the timeline to set so huge a delay? Technical problems? Regulatory hurdles? Locals resisting? Running out of money?

Most of the Hinkley Point C and Sizewell C costs come from interests being paid in the high risk debt raised to build it.

Unsurprisingly it turns out financing such a long term project on a 10% rate debt is not a good idea. (Keep in mind that this debt has been borrowed during ZIRP, where the baseline interest rates where near 0, this rate is just the premium that private actors are taking for the risk to see such a project being canceled for political reason). Of course the sensible way to do it would have been to finance it on long term public debt guaranteed by the British govt, who would then have born the burden of an hypothetical cancelation of the projects (and the cost per MWh would have been 4 times lower IIRC), but public is a forbidden word for the Tories.

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.