Welcome to HCM.
I can very well understand your concern for your mother's health.
Basically she should get a chest x-ray done and her blood should be checked for eosinophilia.
is a very common cause of dry cough
in this part of the world and if diagnosed correctly, proper medicines are available to treat the condition.
Also get an E.N.T. surgeon to exclude any irritation causing pathology in the throat region.
I think proper systematic investigation should help her in relieving her chronic cough
Wishing her all the best.