Can I use my UK MAC desktop in the Cayman Islands
Without knowing the computer you're using it's hard to say, but probably Yes, there should be no problem. 

Most Apple products have an auto-switching power supply which monitors the incoming voltage and automatically switches itself accordingly, others don't, and must be changed manually, and others, are regional specific. 

Here you can find a list of the Apple products and whether if they've got this auto-switching feature.

Checking your computer's manual should shed some light on this, too.