Very thoughtful of you to be considerate of your grandmother. It is very unlikely that her bronchitis will be worsened by contact with the cat, unless she just happens to be highly allergic to cat.
Good luck with your grandma and your cat.
If she is allergic to cats it will, but ask her if she is , and she could ask her Doctor what he thought, will it be in the same house as your Granma,get some answers if she is allergic,