Is it permissible for a woman suffering from a severe cold and cough, who has finished her menstrual period, to perform tayammum instead of ghusl (ritual bathing) with water, fearing that the illness will worsen and that she will be harmed by the cold weather and water?

If you are certainly or predominantly harmed by using water, even if you heat it, then it is permissible for you to perform tayammum and be allowed to pray, touch the Mus'haf, and do other acts for which purity is a condition, until the harm ceases. When the illness ends and the harm is gone, you must perform ghusl.