Im 23w now. I had the same thing a month ago. Went to GP twice and cough didn’t get better within a week. Went to gynae too and was put on another type of cough meds. Most likely mild bronchitis and doctors say it could last a few weeks. Mine went away in around 3. Just rest well, hydrate more and continue taking the meds. It will pass!
Drink plenty of water with lemon or chamomile tea, get prospan cough syrup instead, put a vicks at ur back, throat and lay sleep side ways.