How long should a cold and cough last in a one year old before taking them to the doctor? And can they do anything for the cold if you do take them?

7-10 Days. Many one year olds with cough with a cold for a week or so. If the coughs is severe (disrupting sleep, associated with fever >102, or limiting activity) i would go to the pediatrician. No over the counter medications work that well in this age group. Your pediatrician may prescribe antibiotics for an ear infection, bronchitis, or pneumonia; or a trial of an inhaled medication for wheeze.
2 weeks. After two weeks it may be more dangerous than a regular "flu". Do not hesitate to check with your doctor.