The Holy Land has a relatively warm climate -- they grow bananas and oranges in Israel, including in Galilee (I've seen them myself). Of course, they can have cold rainy days, or very rarely, snowy ones, in December, too. So it could be too cold to have sheep in the fields in December, but not certainly.
The "bleak midwinter" with water "hard as stone" of the Christmas carol is talking about English weather, not Judean weather.
As someone who raised sheep, the idea that ANY TIME of year in Israel is too cold for sheep is amusing.