When does a baby ”inherit” judaism from their mother?

My great grandmother was Jewish, I don’t think she had a Bat Mitzvah, if that is of relevance. My grandmother has never practiced Judaism as far as I know, and is not a religious woman. Would my father still have ”inherited” Judaism?

Google searches haven’t yielded a clear answer, only that Judaism gets passed down from a Jewish mother to her children. I know nothing of this subject, but would love to learn how it works!