In the Peaks, surely the main difference from summer to winter is you go from riding on dry rocks, to wet ones? So anything with a soft enough tread compound should be fine…
For places where winter means wet mud, then look at Maxxis Beavers, Spesh Storm Controls, Bontrager Mud XR and the like. Or for something more all round then Spesh Purgatorys, Maxxis HR2, Maxxis Ikon (you’d be surprised to look at them but they work well in the wet on the back), Bontrager 29-4, Hutchinson Toro, Continental MK2 Black Chilli etc. There’s a tyre I’m particularly keen to try out at the moment, called the Rubena Kratos. Labelled a 2.25″ but apparently measures more like a 2.1-2.15, is tubeless compatible, and has received great reviews from American and Aussie websites especially for wetter weather use. Tread pattern looks aggressive enough without being overly so, and its a dual compound with a high TPI too so should be good.