This is a good point, but as far as I'm aware there is no legislation in the US prohibiting power companies from building nuclear plants right? In fact, there are subsidies and loan guarantees put in place by the government to encourage investment in nuclear. What precisely are oil and coal companies doing to prevent the construction of nuclear power plants?

When you're lobbying to destroy a competitor, it's hard to just pass a law explicitly prohibiting their industry. Media outlets will start asking why, and "because it causes coal companies to lose business" isn't a PR-compatible answer.

So the first thing you do is convince everybody that it's dangerous. Equate power generation with bombs, make a big deal about radioactive waste that lasts for thousands of years as if that's more problematic than ordinary chemical waste which lasts indefinitely, that sort of thing.

Once you've got everybody good and scared, demand safety rules. Paranoid, highly bureaucratic rules. Rules with much higher standards than the rules we use for other industries with a similar risk profile. As much red tape as possible. Make it so you can't unclog a toilet without an engineering study supervised by a team of attorneys. Make everything as arduous and expensive as possible and if anybody objects, accuse them of compromising safety.

Then tell everybody that we shouldn't build nuclear because it costs too much.