Reactive airways refers to bronchial tubes that constrict, in response to smaller doses of allergens or irritants, that do not cause constriction in persons whose airways are not hyperreactive. This tendency of constriction or obstruction is characteristic of Asthma. What you describe suggests that you have seasonal asthma. The radiation therapy may have played a part in bringing this on, but there is no way to know.
The seasonality of your symptoms suggest that you have asthma and not some other damage that could have been caused by radiation. You should check again with your Allergist about the possibility that this is caused by an in-the-home irritant or to an allergen for which you have not been tested. The first thing should be to confirm that you do have asthma and then, if you remain symptomatic despite the use of your current inhaler, for your doctor to step-up the therapy which may only be needed from December until the end of Spring, to control the asthma.
If you work outside the home, another possibility is that your lungs are responding to an allergen or irritant, seasonally present in the workplace.