McDonald's has their Festive Menu and Burger King has this: the Merry Cheesemas menu.
Here's what you'll find on the Merry Cheesemas menu:
Chicken Tendercrisp Extra Cheese - The standard Tendercrisp sandwich we have here in the States but with the addition of Red Leicester and Irish Mature Cheddar cheeses, crispy-fried onion strings, and a peppery steakhouse sauce.
Angus Extra Cheese - A mirror image of the Chicken Tendercrisp Extra Cheese except with an Aberdeen-Angus beef patty.
Emmental Cheese Bites - breaded and fried creamy pieces of Emmental (Swiss) cheese served with a side of red berry (lingonberry) dip.
For the non-cheesy, you'll find:
Snack Box - choose from 20 Chicken Nuggets, 20 Chili Cheese Bites, or 10 of each. Both selections are part of Burger King UK's regular menu. They're just typically sold in orders of 6 and 9. The Chili Cheese Bites aren't something we have in the U.S. but seem similar to jalapeno poppers.
And for dessert there are three, and they're collectively called "BK Hotties" (I don't think "hottie" has the same meaning in the UK as it does here):
Hot Raspberry and Chocolate Blondie - A blondie is like a brownie but with brown sugar instead of chocolate with a taste sometimes describe as "like butterscotch." The center of this one contains hot raspberry and white chocolate. It's topped with soft serve.
Hot Toffee Sponge - A toffee sponge cake soaked in toffee sauce with a bit of toffee sauce in the middle, topped with soft serve ice cream.
Hot Chocolate Brownie - A dark chocolate brownie with a gooey hot chocolate center. Also topped with soft serve.