I'm beginning to loathe the "being in the rain causes characters to fall deathly ill with a cold" trope. First of all, rain doesn't cause colds. Secondly, colds aren't all that bad. If you have a deathly fever, it isn't a cold.
Depends on your body and where you live, tbh. I'm over in Ohio (America, but a part where the seasons can be unpredictable and rough), and for me personally, a cold can knock me on my ass flat for nearly a week sometimes. Rain doesn't directly affect colds, no, but depending on your immune system, sudden shifts in the environment around you (going from a quick 80ish to high 60s with rain or high humidity as an example) can kick your ass if the germy-germs in your body see an opportunity. Nearly double that chance if you deal with allergies.