Doc says I have chest infection n gave anti biotics. How soon can I expect my shortness of breath to be relieved. He says I dun have bronchitis.?

Monitor closely. If your dr is confident of the diagnosis of bronchitis ( no pulmonary emboli or asthma/ copd, pneumonia)the "shortness of breath" should improve gradually while the cough is disappearing. It may take days or weeks, seldom 1 month or more. Shortness of breath should be monitored closely to make sure there is no serious condition causing the difficulty breathing.
Probable time..,, If the antibiotics are effective for the infection you have, your symptoms should begin to improve within a few days even though it may take a week or 2 for the infection to completely resolve. If your symptoms are worsening despite the antibiotic therapy, then you need to be re-seen by your doctor or in the er, depending on the severity of your symptoms.