No, that is heavily sensationalized. But a new 'small ice age' as a result of the gulfstream shutting down is a serious possibility, even if it changed direction that would already cause serious problems for very large parts of Europe.
Although the benefit would be increased reflection of sunlight due to more icy (reflective) surface which would alleviate (or reverse?) the current weather warming trend.
The basic premise is faulty, a chance in the ocean currents would not immediately cause massive climate change, it would take a couple of centuries but would be pretty much inevitable. It likely would also accelerate other processes already in motion, so more extreme weather would be a pretty safe bet but not to the degree depicted in the movie.
Well, they forgot to account for adiabatic heating of that cold air getting pulled out of the upper atmosphere, which would have made it quite hot when it hit the ground. If I recall, that cold air was the central threat in the movie.
We actually ran the numbers on that as an exercise during my undergrad degree (meteorology), iirc it would end up the same as the surface air temperature
