while i don't disagree that it will take time, i feel it will depend on the location of the port and will vary by cruise line. Royal Caribbean specifically is well positioned to sail almost immediately since they're headquarters in Miami and a lot of staff remained local when the lockdown began. RC also has two private island options they can sail to nearby that don't require special permission from local government due to covid red tape for all the precautions you mentioned above, and i suppose some bribes, sorry, i mean expedite fees.
Perfect day at Coco Cay is an amazing destination with lots to do so even short trips are very viable for RC.