This is very good, but you need more than verbal undertakings. You need paper, in particular you need a Medical Power of Attorney, a notarized document authorizing you to make medical decisions on behalf of your Mother and Grandmother (and your Mother, for the Grandmother) (oh, and mutually reciprocal ones for you and your spouse if any).

Without that paper, hospital personnel may ignore you when you say what she wanted, in fact may feel they are legally required to ignore you.

And the paper needs to be accessible on short notice (not in a bank safety-deposit box). It's a good idea to have a copy filed at her primary care doctor's office.