Some common causes of chronic cough include asthma, allergic rhinitis, sinus problems (for example sinus infection),post nasal drip and esophageal
reflux of stomach contents. . Infections such as bronchitis or pneumonia can also cause coughing. These infections can be caused by virus, bacteria or fungus.
In your case, as you are having constant nasal blockage, so you could be suffering from chronic sinusitis leading to cough.