Very interesting. I go through phases of coughing after cardio exercise and only now realize the correlation with the seasons!
Though late last winter I made a conscious and extremely difficult effort to exclusively nose breath while doing my 5 km runs, inspired by a slightly fuzzy pop-science book. I experienced much less coughing after exercising in the cold after this change. I believe it has helped all my cardio activities since making this change. It was difficult because early on I would get an almost completely blocked nose due to the cold air, and instead of reverting my breathing behavior I would be forced to stop and wait for it to clear. After a while this issue disappeared.
This gives me hope. I thought I just couldn’t breathe through my nose while exercising. I find it super uncomfortable to unbearable when I feel like I’m not getting enough air. I didn’t realize you could just practice more.
I had the exact same experience as you at first - "there's no way this is going to work" - but two things helped, and now I can breathe through my nose doing an hour of cardio no problem (and I feel very calm in fact - it's pretty great).
The first thing I did was that at the start of the workout, when breathing through my nose and immediately getting the sensation that I wasn't getting enough air - I'd force the air through my nose really hard - like _really_ hard - to the point where sometimes I'd get a bit of mucus (as if you were blowing your nose). You'll be surprised how much air can pass through your nose if you push it in and out quite hard.
The second thing was that I'd just start slow. I wouldn't start with a sprint, instead I'd slowly warm up, focusing on my breathing through my nose (with the first minute or two always being that forced push/pull mentioned above). Once my breathing and heart rate felt stable (I track my heart rate whilst I exercise) I would then gradually pick up the pace.
You'd be darn surprised how fast you can go / how high a heart rate you can sustain whilst breathing through your nose. I've sustained 150 bpm for over an hour whilst comfortably nose breathing. It just takes practise. And a bit of a messy nose at first :)
Though late last winter I made a conscious and extremely difficult effort to exclusively nose breath while doing my 5 km runs, inspired by a slightly fuzzy pop-science book. I experienced much less coughing after exercising in the cold after this change. I believe it has helped all my cardio activities since making this change. It was difficult because early on I would get an almost completely blocked nose due to the cold air, and instead of reverting my breathing behavior I would be forced to stop and wait for it to clear. After a while this issue disappeared.