First, there is no evidence that the Gulf Stream will cut off. The Gulf Stream is being measured, but it’s a fairly new science. There could be some truth in the suggestion that the salinity of the sea being undermined by the melting of the ice caps. That would interfere the global mechanism that transports warmer water northwards from the equator towards us, where it cools down, becomes denser and returns southwards in the deeper oceans. But as to whether it would happen, that’s still pure conjecture.
"Continent-type weather sees more extremes, so if the Gulf Stream stopped for some reason Britain would become colder, with more snow."
But currently the Gulf Stream, which is what we call that warm ocean current, keeps the UK and Western Europe that much warmer than it would otherwise beat our latitude. If you compare us to parallel places such as Newfoundland and New York, they get more continental-type weather - by which I mean the sun bakes the open space far more than it does an island such as the UK. When you’re surrounded by sea, it creates a kind of moderation.
https://www.youtube.com/embed/UuGrBhK2c7U?wmode=opaque
Continent-type weather sees more extremes, so if the Gulf Stream stopped for some reason Britain would become colder, with more snow. The model would suggest that we would start to inherit more of that weather. The sea would start to cool in and around our shores, and the temperature will drop. It will be drier too. This would be a long-term event if it did happen. It wouldn’t happen overnight. But with global warming, who is to say when it might?