Weather could be a challenge. But you might not mind whatever the weather has in mind. A bigger problem I could imagine would be difficulty in finding lodging. Best to research whether albergues are even open during the Christmas holidays. Also, other lodging options could be limited, as Spaniards like to enjoy the holiday season and some establishments might be completely closed.
One last consideration would be the state of some parts of the trail. There are alternatives, but I recall one day in particular, just west of Laredo, where we had to climb a very steep hill where the path was rutted and all slippery mud, plus there were no handholds to help pull us up. And without handholds, not even a branch or a root, we could not make it up the incline but instead had to backtrack and take a different route. At our lodging that evening the proprietor told us about a pilgrim who, the day before, had managed to get over that part which was followed by,
I think, something like 6km walking in sand. The end result was the person was so exhausted and dehydrated they had to send them to the hospital in an ambulance. All of which is to say, there are a few challenging spots that, although ok in good weather, could be problematic in winter rain, especially when there surely won’t be many (if any) other pilgrims around.
Sorry, I don’t mean to be discouraging! One year in late February, my husband and I flew to Spain thinking we would walk the Norte. We first visited relatives in Alicante. When we saw a weather report about terrible storms in the north, we abandoned our plan and walked from Alicante instead.
The route is beautiful though it could definitely present more challenges in winter.