As f-mattox stated, it's up to the Captain. You're going there when the east coast of the US is on Standard Time (as opposed to Daylight Time). Not sure if this makes a difference. I do know, this time of the year, when the east coast is on Daylight time, Grand Cayman is one hour behind us. I'm not sure if it's because they don't observe Daylight Saving Time or whether it's because they're on what the United States refers to as Central Time.
On my last cruise with Royal Caribbean, the captain changed the time to match local island time, but on my cruise coming up, the captain probably will not change the time from what I've been researching.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)