All British citizens must travel with proof of citizenship, which is a valid passport, or birth certificate (complete with raised seal, not a photocopy) with a picture identification, or a certified naturalization certificate with picture identification. Any state government issued drivers license or other form of picture identification should be acceptable on domestic cruises that do not travel into international waters.
Non-UK citizens who are residing in the United Kingdom are required to travel with their valid passports. All must their necessary visas at embarkation. You should consult with your travel agent for more information if necessary.