Some would say it is because of the influence of Charles Dickens, with particular reference to A Christmas Carol. Ebenezer Scrooge's office is like a refrigerator and the old skinflint's apartment isn't much better. The truth of the matter is that the Christmas of the year before the story was written was the occasion of an exceptionally heavy snowfall and Dickens no doubt saw the dramatic possibilities of such inclement weather. Also, coincidentally, the Christmases of the first ten years of Dickens' life were white Christmases. His childhood memories of Christmas would have involved snow and snow would probably always have been in his image of the Season.
Another suggestion is that, in England, it quite often snows in January. But when Europe switched to the Gregorian Calendar, England continued to use the Julian Calendar until the 18th Century. This meant that Christmas Day fell on the day now called January 6th and the likelihood of snow on this date was somewhat greater.