Only South Korea, Russia, and China appear to be able to build nuclear plants on a reasonable budget and timescale. And I'd put an asterisk next to project costs in Russia and China because they don't have great domestic transparency and the projects they offer to build abroad are pretty expensive.

In addition to high costs, nuclear power projects often face opposition from the public in developed democratic nations.

The region with highest current demand for desalination, Middle East/North Africa, is "blessed" with a lack of democracy. If the Saudi monarchy wants nuclear power they don't have to get voters on board -- no voting! But the costs are still high. And although Saudi Arabia would like to decrease its domestic fossil fuel consumption to free up more for exports, it doesn't want to accelerate a shift away from fossils elsewhere. IMO it's kind of a tightrope walk between advancing their own non-fossil energy sources, to grow/preserve export revenue, and avoiding a global shift away from fossils that could diminish their export revenue.

I've always thought it really cool how we can collapse the water problem into the energy problem. (And, to be clear, I agree with your solution to the energy problem.)

Because it's expensive, has public stigma, and some places like California are located near faults.

Nuclear is more costly than vast majority of other energy sources. Solar makes more sense if you want to stay green, and if you don't care about the green aspect traditional energy generation is probably better.