In reply to Bob:
Yes I do agree with this. If you are chesty/coughing up stuff, you probably shouldn't go running, especially in cold air for a prolonged period. I guess you might get away with the treadmill for 20 mins, but really better to rest completely, as frustrating as that might be.
I didn't do this and ended up prolonging (I believe, though for all I know this could have happened anyway) a chesty cough (in fact, it got worse, before finally shifting a couple of weeks later).
Now, despite my advice, I would probably do the same again, because I'm a stubborn idiot.