Even if current weakens it won't make Europe cold. Main effect will be extreme hot subtropical ocean and stronger hurricanes in Caribbean and south US. Those hotter sea levels will still distribute heat along the main current path. But Europe will get warmer because Atlantic gets warmer without current. Hot water spreads naturally, and hot air temperatures make water warmer. Continuous dropping of Artic ice levels, especially the winter maximums, means very long summer warming periods and very high fall arctic ocean temps, means steady ocean temperatures at UK level.

AMOC collapse theory is based on accelerated Artic melting. But what's happening so far, last decade or so, is less freezing happening each year. At the time the theory was developed, global warming was led by Arctic region, warming 3-4x faster than rest of planet. Rest of planet has caught up without maintaining that ratio.