Welcome to HCM,
After going through your history, you had chest infection
that is bronchitis
which may be secondary to viral infection or if you smoke your lungs may be inflamed to develop infection.
There is delay in initiation of treatment but this is good you have improved partially,if only cough is there it will improve in few week time by symptomatic treatment.
If there is residual bronchospasm
drugs like bronchodialator and anti inflammatory therapy may be required.
If already X ray
chest is not done it may be done along with routine blood counts.
If you smoke please quit smoking
which will help a lot.