Citation needed. A quick look at Google seems to contradict this. In fact, I always thought the GS made it to Britain and North England. That said, let's not derail this discussion: just a link would suffice.
The best article I know of is this:
It makes clear that it is not the water from the Gulf Stream that warms Europe, bur rather the Gulf Stream and its downslream components (see below) that warms the air masses above, and it is those air masses that get to Europe and keep Europe warm.
The Wikipedia article is not bad.
en.wikipedia.org
See the section on "Properties."
The confusion often comes from the fact that the Gulf Stream is really just one part of the circulation in the North Atlantic; when it gets to south of Newfoundland it breaks up and feeds the North Atlantic Current (going northeast) and the Canary Current (going southeast). It is the North Atlantic Current that warms the UK and Europe, not the Gulf Stream.
As the Encyclopedia Brittanica puts it, "The main portion of the Gulf Stream continues north, veering more to the east and passing close to the Grand Banks, south of Newfoundland, where it breaks up into swirling currents. Some of these eddies flow toward the
British Isles and the Norwegian seas and form the
North Atlantic Current (or Drift). A larger number flow south and east, either becoming part of westward-flowing countercurrents or joining the
Canary Current."
The Gulf Stream and the North Atlantic Current are driven by very different forcing dynamics; the GS is a west-ward intensification of the Atlantic waters driven by windstress, primarily to the west in mid-latitudes. The North Atlantic Current is driven by thermohaline processes (rising and sinking of warm and cool and fresh and salty) water.