Hi,
Welcome to ehealthforum. From the description of your symptoms, it seems likely that the cough is due to bronchitis. Apart from antibiotics, ask your partner to take plenty of fluids throughout the day and inhale steam as many times as possible.
Inhaling steam from a bowl of hot (but not boiling water) may help to soften, and loosen, the build up of mucus in the nose. You may also find that adding some menthol crystals, or eucalyptus oil, to the water, helps to ease your symptoms. He can also take antihistaminics like Benadryl or zyrtec for some days and see if they provide relief or not.
Some chronic cases may need oral steroids or steroid nasal sprays and bronchodilators. These can be taken only after prescription.
I sincerely hope that helps. Take care.