I have always heard that most of the iconography of the modern Christmas (not just the snow, but robins, holly, mistletoe, and the like) comes from Victorian England, and it does (sometimes) snow in England at Christmas. Charles Dickens’ A Christmas Carol (and his other, now lesser known, Christmas stories) had a lot to do with how we think Christmas ought to be (and DIckens was very popular in America, and elsewhere, not just in Britain). In England these days, it rarely snows until well into January, but in Dickens’ time Europe was still in the grip of the Little Ice Age, and snow would have been common and often thick in December. I do not find it at all hard to believe that white Rhodesians, especially, would have been anxious to have as British a celebration of Christmas as possible.
Christmas trees (and perhaps other aspects of the Victorian British Christmas) were supposedly introduced into Britain from Germany by Prince Albert, Victoria’s husband. In Germany, fir tress like that are, I think, particularly associated with the Black Forest, which is quite high ground, and so also probably often snowy in December (especially during the Little Ice Age, but perhaps even now). Thus the German Christmas may traditionally be snowy too.
ETA: Darn! Semi-ninjaed by casdave!