John's grandmother / She

If you have not introduced any other females, she is alright.

So, if you write: John's grandmother's birthday is a week away. She wants to make a new dress for herself. --everyone will understand whom you are talking about.

But if the sentences are separated, and there are other women you are writing about, you might want to specify who "she" is by naming John's Grandmother again. If the two sentences are consecutive, "she" is fine.

As I understand it, you are asking whether "John's Grandmother" or "She" is prefereable in the second sentence, if that second sentence immediately follows the first. Definitely use "she." Repeating "John's Grandmother" sounds awkward and repetitious, unless you're trying to achieve a particular effect by referring to the lady as "John's Grandmother" every time she is mentioned. But that would be a particular literary device. In general, where there can be no ambiguity, use the personal pronoun in the second sentence.