Which ocean current's failure is the mechanism behind abrupt northern European cooling?

The Atlantic Meridional Overturning Circulation - the AMOC, of which the Gulf Stream is the visible part. Warm surface water moves north, gives up its heat, becomes cold and salty enough to sink, and returns south at depth. The sinking is the engine.

Fresh water stalls it. Meltwater from Greenland and increased rainfall dilute the surface, which stops it becoming dense enough to sink, which slows the whole loop. The geological record shows this happening abruptly at the end of the last ice age, and Greenland ice cores record temperature changes of several degrees within a decade - which is the alarming part: the system appears to have two states and to switch rather than drift.

How close it is now is unresolved. Some analyses of sea-surface fingerprints have suggested a collapse could come this century; others find the observational record too short to say, since direct measurement only began in 2004. What is agreed is that a shutdown would cool north-west Europe substantially while raising sea level on the American east coast, and that the film version, with a wall of ice arriving in a week, is not what anyone is describing.