Everywhere in the Caribbean will acept US dollars cash. If you offer Euros cash anywhere other than the French/Dutch islands they will normally refuse it (I'm not even sure some of them know what it is).

In our experience every non-European island has a fixed rate against the dollar. ATM's are no problem for UK cards, whether they are for pounds, dollar or euro acounts. A few island ATM's will offer dollars or local currency (it's not normal though). If you are moving on quickly (and not going back) take dollars if you can.

If you move around islands expect the card(s) to be stopped occasionaly. The banks aren't used to people moving around much. After a year or two they get the message.

If you head to Venezuela definitely take dollars, the black market exchange rate is roughly twice the official rate.