I think we would see it in stages like we have with cars. We already have track steering where the ship self adjusts heading and propulsion.
Next will come assisted navigation where the computers suggest a course and speed for traffic avoidance and route planning. No more trial maneuvers, the computer would offer maneuvering options. The human would still approve or reject the choices. Think navigation systems in cars.
Then comes extremus avoidance where the ship would take over to avoid collision or grounding. A human would have to explicitly disapprove to prevent an autonomic action. Some cars do this now with breaking and lane changing.
After that it would be automatic navigation where the computer would do it all and request help only when it can't solve a problem. The thing about this is ships, like cars and planes, would likely be safer then if they were operated by fatigued or inexperienced humans.
And finally, only after many years, the dangerous, unreliable and error prone humans would be removed completely and a fleet of ships would be navigated safely from a control center in a cheap Third World country.