We could very well be heading towards 'the day after tomorrow' but not anywhere near the scale the movie depicts it as. Its virtually impossible for the global CLIMATE to change so quickly. Freak weather patterns, yep, storms, yep, but the world's entire climate cannot possibly change so quickly. The geography of the movie is at fault, also; it describes massive storms forming over large land masses. Storms of that type particularly cannot form over land masses so large and dry.
Also, it would require our polar ice caps to melt at a much faster rate. Cold, freshwater from the ice forces the WARM salty water below it. The warm current would be deflected away from places such as the UK, giving it a climate like Alaska or something along those lines, probably cooling Canada and the northern states of america... but the sahara desert would be likely to expand to, into Spain... Basically, everything would just go to extremes. But that's not gonna happen all at once, like in just one year... but over time, however short...
But, let me put it into perspective. By 2100, global average surface temperature could be anywhere between 1.4 and 5.8 degrees higher. We better pray to whatever gods we believe in that its NOT the latter - a rise of 4 degrees caused the last major ice age 14 thousand years ago. Theoretically, the earth within a few human generations could be hotter than at any time during the last 50 million years...which in turn will probably increase the rate of species extinction...
Humans haven't tamed the earth, oh hell no... we've given mother nature the ammunition to be more 'violent' than ever before, so to speak...
You guys will have to excuse me LOL. I can be very passionate about the idea of climate change and the destruction of unique ecosystems... I could live, eat, sleep and breathe meteorology, too.
As fr the time thing... that's an interesting question. I think it depends on how much faster the earth is spinning now... it could be such a negligible amount that it doesn't need to even be thought about, I guess...
*Edited to add: On the topic of freak weather systems, I'm quite excited! I do believe that two days ago I bore witness to ball lightning which dissipated upon contact with the road surface near me. ball lightning is a rare enough occurance already, but to see it near me - privilage!