Naval reactors are expensive upfront and expensive to operate. So the likeliest path to commercial nuclear vessels is onerous carbon regulations. That path isn't likely.
Naval reactors have some design choices that don't make sense for shipping. Things like they can shutdown and restart the reactor quickly. A Submarine sometimes needs to get into max quiet mode as quickly as possible and then when the threat is gone get back up to full speed. This is obviously required for a war vessel, but for a commercial ship can use a cheaper reactor that can't be abused like that.