Not even remotely accurate. It would cause notable changes and likely widespread famine but not anything as instantly apocalyptic as the film. Temperatures wouldn’t change dramatically aside from Northern Europe. The gulf stream’s effect on temperatures is smaller than people think.
The bigger change globally speaking would be shifting precipitation patterns. This could lead to dramatic aridification of parts of Africa and India, and large increase in precipitation in North America. But it’s not going to trigger some kind of hemispheric ice age as depicted in the film.