As a Mongolian citizen planning to travel on a cruise, do I need a visa to each country or island where the cruiseship stops? If I already have a visa to the UK, will that be accepted in any country/island that does not require a visa of a UK citizen?
will that be accepted in any country/island that does not require a visa of a UK citizen?
A visa to the UK does not entitle you to be treated as a UK citizen. A British possession (or any other jurisdiction) may choose to accept a British visa as excusing certain travelers from any requirement they would otherwise be subject to to have a visa, but this will vary from place to place, and probably also depending on the type of visa. It's therefore not possible to answer the question generally.
It is not uncommon for countries that receive cruise vessels to have exceptional arrangements for the passengers on those vessels, allowing them to enter with different documentary requirements than those to which they would otherwise be subjected. You should therefore ask the cruise company about the documentary requirements you will face, and you should also check with each country where the ship will be calling to confirm the information you've received from the cruise line.