Can I use my UK MAC desktop in the Cayman Islands I am moving to the Cayman Islands, which uses the same power source as the US, will I be able to use my MAC desktop there using a US adapter? 

Thanks
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.