It can be quite common for a cough to persist for a while, after recovering from bronchitis. As for the next steps to take, you may want to check out these 2 similar questions that were previously answered by our colleagues here:
How do I get rid of a persistent cough?
What could be the cause of a persistent dry cough?
What could be the cause of chronic cough at night and in early morning? How can I stop the cough?
In most instances, your cough will gradually get better over time, with some of the symptomatic medication that your GPs would have prescribed.
However, if it is still severely bothering you, I'd suggest for you to pay a visit to an ENT specialist who can look for other possible triggers for your cough, as covered in their answers above. Hope that helps!
