When should I take a coughing child to see a doctor?

If persists... Any cough lasting for more than a couple of weeks needs to be evaluated. Acute bronchitis of either viral or bacterial origin typically resolves within a few days to a week or so. Therefore, if the cough lasts longer than that or if it is worsening, then it should be evaluated by a doctor.
Watch for distress. Coughing by itself is not worrisome. However, for persistent cough lasting greater than several days, or other warning signs such as difficulty breathing/respiratory distress should prompt medical attention.