How would you predict the climate to change on Islands if the Gulf Stream did not exist?

In the Earth’s past, scientists have seen evidence of large inputs of fresh water into the North Atlantic from melting glaciers and ice caps as well as changes in the thermohaline circulation during transitions in and out of glacial periods. Global warming could potentially cause a thermohaline circulation shutdown and subsequent regional cooling, but because Earth will continue to warm as a result of greenhouse gas emissions, it would not produce another Ice Age. If the thermohaline circulation shut down, cooling would likely occur only in regions that are currently warmed by the ocean conveyor. And even if the thermohaline circulation did shut down, winds would still likely drive the Gulf Stream; however, there would be less warm water from the tropics and the Gulf Stream could become cooler and not reach as far north