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.
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.