If you are flying direct (not through USA) then no visas are needed but you will need to take your passport with you whenever you leave the ship as well as your sea pass card to get through security etc.
As you are only visiting via a cruise ship no normally, on some islands you may have to carry your passport with you or at worst a photo ID. I certainly have visited all these islands and not had a problem so my advice is purely to be on the safe side. I wish I was going with you our next Caribbean visit not until the spring. One piece of security advice never give the date or the last two weeks in... as it might be picked up by someone who would like to know your house might be empty, we don't know who is looking into this website, doubt there is a real problem but be careful in the future.
Royal Caribbean - Independence of the Seas - 14 Oct 17 - 7nts
Southampton : Gijon : Bilbao : Cherbourg : Paris (Le Havre) : Southampton
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Southampton : Lisbon : Valencia : Cartagena : Gibraltar : Seville (Cadiz) : Southampton
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Princess Cruises - Caribbean Princess - 20 Apr 17 - 11nts
Southampton : Guernsey : Cork : Dublin : Hamburg : Rotterdam : Brussels : Paris : Southampton
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