For me, if the snot turns green (=infection) then we go, if their chest sounds rattly (you know that sound when you breathe and you have bronchitis?) or if the cough has been chesty for a certain number of days (the younger bub is, the shorter I leave it) without sounding better then we go. I just monitor the situation really closely, and if I feel it's getting worse (especially if it gets worse over a short period of time), then I act.

As an example, both my boys have had chesty coughs the last week (week and a half for the eldest). I kept them warm (not hard on the hot days!), their mucous was clear, kept them away from other kids, and blew their nose lots, was happy with the amount of coughing they were doing (and where that coughed up mucous was going, by checking the nappies...ew), and they didn't get rapidly worse, and then the cough started to clear up for them after a few days. Most importantly, I didn't feel concerned.

If you're concerned at all, don't ignore that, take bub to the doc, who can listen to their chest (take them when the cough is worse, like late arvo or early morning). The worst that can happen is that you'll feel you could have spent the time better when the doc says 'it's a virus, nothing you can do, keep them warm, head elevated, humidifier if trouble breathing, blowing nose, come back if it gets worse'.
By the sound of the nappies, at least some mucous is leaving his body . But go to a doc if things don't seem right to you.