Re the Conveyor: Takes a thousand years over all, a lot of it in the cold deep, but the warm surface current travels much faster, as I recall...which is not the big problem.
The main problem is the meltwater coming into it out of Canada, which is where fresh water gets into the act---that and the fact that the ice has broken up in the north pole regions, and there's open water up there---ice is freshwater, pure and simple, and it's melted into the mix. Not to mention the major pieces that have come off the south polar ice sheet, which will also melt as fresh water when they get to temperate seas. If the salinity changes due to too much fresh water, the theory goes, the Conveyor stops. Research has been studying ice corings and such going back multiple thousands of years, figuring out the weather patterns and how it coordinates with local geology, etc.
Indications are when this process of salinity change starts, the shutdown of the Conveyor is fast, and the warm water no longer sweeps past the coast of England and the US West Coast. Which means we up in the north get out the snowshovels and people in the southern US prepare to bake. Heat pools where it tends to be hot and cold where it's cold, because the thing that moves it on around the world has switched off.
If that thing does switch off, you got to pack up and get north, is all, Grendl and Phil, and help us Spokanites shovel the white stuff.