The Windward Islands use the same electricity supply (230V at 50Hz as the UK does). They even use the same mains plugs and sockets. So there would be no problem with plugging in the base unit.
However it's worth remembering that the frequencies allocated to radio devices vary across the world. A cordless phone designed for use in the UK might well use a frequency which is reserved for another purpose in Grenada. If their frequency allocations are different to ours the phone would still work (as long as it was used with its UK base unit) but it might pick up interference from other services (such as taxis, public utilities, mobile phones or even the military). Worse still, it could cause interference to such services, leading to prosecution for using an unlicensed radio device.
As an analogy, it's strictly illegal to use most American radio devices (such as cordless phones, walkie-talkies, etc) in the UK, as they use frequencies which aren't authorised for such use here, or otherwise fail to comply with our regulatory requirements (e.g. in regard to the power levels in use). Attempting to use such a device here would lead to it being seized and probably to prosecution for what is regarded as a serious criminal offence.
Chris