This is known as "pink hydrogen." There are already many proposals to do this: https://www.azocleantech.com/article.aspx?ArticleID=1718
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Thanks! So I'm not the only one thinking this, but it's held back by high initial capital costs and public perception that nuclear power is dangerous (thanks fossil fuel assholes).
FYI for anyone who doesn't want to read the article, there are four nuclear power plants gearing up to produce pink hydrogen in the US, and a project in Sweden.
They conclude that "there is growing anticipation that nuclear-enabled hydrogen projects will gain momentum as governments strive to enhance their nuclear energy capacity for a green transition, resulting in pink hydrogen emerging as a more efficient, cost-effective, and sustainable alternative to fossil fuels."