I read these lines in a novel recently
"It was one of those mild indefinite days, which we now associate with Christmas in England, a day which had in its clouded sunlight nothing of autumn and nothing of winter, but seemed rather to suggest the spring — a feeble spring seen in a looking-glass. ... These meteorological reflections of mine were prompted entirely by the feeling that I was perhaps wearing too thick an overcoat. ..."
Sounds bang up to date? This C H B Kitchin in Crime at Christmas first published in 1935!
Apparently, the idea of Christmas being snowy can be traced back to the fact that eight of Charles Dickens' first Christmases were white. It is suggested that he associated snow and Christmas in his successful Christmas novellas and we all followed suit.