Nothing will change at the Winter Bash page (https://winterbash2018.stackexchange.com) after Jan 1st. The main WB page and the leaderboard will display as is until the SE team redirects it to the Stack Overflow Contest site later in the year.
The pages will stop updating after Jan 1st since no new hats can be earned and the ranking shown on the leaderboard after then will be your final ranking.
After the whole WB page is removed, you won’t be able to see your ranking (similarly, you aren’t able to see your last year’s WB ranking).
However, hats on your profile will be removed on Jan 1st and you can’t change your hats after that.
There will also be a wrap-up blog post usually published 6-8 days after WB ends. (Past blog posts here: 2017, 2016, 2015, 2014, 2013 & 2012)
So, take lots of screenshots while you still have your hats!