Britian would normally be warmed by ocean currents.
Since it's cold, it indicates the ocean currents are not functioning as they should, and a likely cause is that global warming is changing the temperatures and salinity of the water to the point where the currents no longer function normally.
Why wouldn't global cooling do the same?
It would affect temperature, but not salinity.
Salinity is affected by themelting of various ice shelfs, which would cause the water to be less saline.
And why would a less saline water be the preferred explanation over a more saline one?
Currents are built on water salinity. The more saline water is heavier, thus making it sink.
At different depths, the water flows in different directions. The salinity changes within the current through regular freezing and melting, thus allowing the water to reach the depth and flow in the directions it needs.
Global cooling would affect the freezing, but negligibly, as said before
Melting has a greater and quicker impact because a lot more water is added to the system via melting than is removed via freezing.
So, with the salinity not at the places it needs to be, the current cannot flow correctly.