Private security is not a guarantee and someone could always send a suicide bomber. It happened a few times to US navy ships it would probably be easier to do to commercial ships.
Four have been built (by the US, Germany, Japan, and Russia), starting in the 1950s (one per decade from the 1950s to the 1980s), but it's never really caught on. The most recent built was refitted and returned to service in 2016, after having been decommissioned and awaiting disposal for nearly a decade, the rest have been scrapped.
Too expensive and complex. Even global navies don't use except in unusual circumstances that absolutely require it, like subs, aircraft carriers, and a handful of polar icebreakers.
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.
I’d expect gas and diesel to be used for airplanes and ships for a long long time.