Speaking for the RN: If there were a situation that called for wearing a full uniform, then yes, officers and petty officers (and Marines) would wear their full English broadcloth uniforms and be tremendously uncomfortable. Ships could rig awnings and pump water on deck, but most of the time people would just have to suffer along with the heat.

Full uniforms would be necessary for some events, including punishment of men, the ceremony of divisions on Sundays, formal dinners, and the like. That said, most of the time, officers and men would be in working clothes that would be adapted to the climate (whether they were in the Caribbean or anywhere else). For men in the Caribbean, that would include loose trousers, long-sleeved shirts and sun hats (made of sennit or duck canvas); officers might wear light jackets, but also would wear loose breeches or even trousers, and would wear shoes (the men generally didn't wear shoes while working).