The big blocked mud tyres like BF All Terrains will do OK in deep dry snow, but for cold frosty tarmac, icy slush and wet snow, they will be terrible.
If you want something with an aggressive tread for mud as well as winter, get something like the Nokian Weatherproof (all-season). If you want an well rounded tyre get the Crossclimate.
Winter tyres work because of the rubber compound and multiple sipes cut into the tyre, offroad tyres do not feature these. I always run all-season tyres but this time I got General Tire Grabber AT’s, I do not expect them to do well in winter at all, but I don’t need it for snow anymore, just summer and mud.