A UK ship is steaming out of the Caribbean through a narrow strait between two island States. What rights of passage, if any, does the ship have?

The right of straits transit passage under UNCLOS Article 38. 

States bordering straits may not impede transit passage and must give appropriate publicity to any navigational dangers. Transit passage cannot be suspended, but coastal States have a limited right to regulate passage through straits. They may establish sea lanes and traffic separation schemes which vessels must observe.