We had a wonderful little filly born last year in Alaska in the snow -- inside a trailer in the middle of the night. I should know, I was on the phone with them.
Baby did just fine, I believe they used a blow dryer to dry her off, and had a nicely bedded 'stall' for her, and she did great. Not to worry, you'll get through this. Remember, many people breed for early January babies, so they are 'mature' for the show ring, so we'll help you prepare, and I'm sure baby will do just fine.