I suspect that the answer will depend on your tour operator and travel agent. In the case of UK based cruise, it is possible that you need only arrange a passport, the booking agent or travel operator will do the rest. You will need to check with your travel agent or the cruise operator for confirmation.
When I went on a Carribean cruise with a well known UK cruise operator they arranged all the vias as part of the booking. I am a UK citizen with a UK passport and I had to make sure I had at least 6 months on the passport left at the time of my 2-week cruise.
At many of the docking point, there were no port staff actively checking passports and vias. The crew of the boat worked with the local authorities to check you on and off the boat in an acceptable manner. The boat operator had either got a waiver or had applied for vias on my behalf. I had to carry my passport on some islands and boat ID card on all.
My cruise started and ended in Barbados. I flew into there from the UK, went through long ID check and security process once then boarded the boat. This process was repeated again at the end of the cruise.