I spent last winter focusing hard to get about an hour of cold exposure daily. I worked out and did intervals on a treadmill in my 40 degree garage and took a walk in below freezing temps nearly every day, dressed minimally. I live in Alaska, so I can do that!
I hate to say it, but there were no amazing results. I was able to maintain my weight over winter--which is hard. But I also think that prolonged cold exposure seems to make subcutaneous fat even more stubborn. This is actually, probably, a good thing otherwise we'd be extinct--but not so good for people looking to manipulate the cold for fat loss.
I have actually had more success with this crazy thing that anything else, cold related: https://www.coolfatburner.com/ I think that wearing this ice vest in an otherwise warm situation causes more calorie burn and fat loss than prolonged exposure to cold temps. I also think that the cold showers are better than freezing your butt off outdoors.
It seems to me that nature has a master plan for us when we are exposed to cold weather and seasons that we cannot escape or manipulate to our advantage. Brief periods of cold, like exercising while wearing a CoolFatBurner or taking a 5-10 minute cold shower and then warming up, cause a rise in muscle contractions and some other metabolic effects that are more advantageous for weightloss than trying to stay cold all winter. I think that once the cold becomes chronic--hormonal changes occur to preserve fat for survival. Just my N=1 in a really cold setting.
I had hoped that prolonged cold exposure would result in very low bodyfat content and give me the ability to eat massive quantities of food without gaining weight--but I experienced the opposite. I even tried a ketogenic diet during that time and saw no difference.
I would say that if you want to increase BAT and get a bit more bang for your buck, periodic cold showers (which I HATE) or wearing an ice vest while working out are the best methods.