Baby could get a runny nose while teething that’s quite common - but as above if they have a cold just bad timing xx
My little girl has a runny nose out of no where at the moment, and no one in our house is sick either! I was actually thinking it might be hay fever, with the pollen count being high at the moment.
Cold symptoms aren't associated with teething. It's probably just bad luck and your baby has a cold at the same time as teething!