All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
The scholars differed in opinion concerning the ruling on awakening the sleeper and alerting him for the prayer. Some of them held that this is recommended, while others held that it is obligatory. In any case, we advise you to awaken your grandmother for the prayer and remind her of Allah and the fact that she should have good expectations of Him. You should try your best to instill the meanings of faith that would help her get rid of her fear in her and be kind to her.
Since you suggested to your mother that you sleep with your grandmother at that time, then if you were to wake her up, it would not harm your mother. Hence, we advise you to wake her up and give her advice and remind her of Allah, as we mentioned above.
Nevertheless, if you do not wake her up, you bear no sin according to the view of many scholars.
An-Nawawi said, "It is recommended to wake up a sleeping person for the prayer, especially if its time is short (i.e. its time will expire soon), as Allah says (what means): {And cooperate in righteousness and piety.} [Quran 5:2]"
Also, ‘Aa’ishah, may Allah be pleased with her, reported that the Prophet used to offer prayer at night while she laid in front of him, and when the Witr prayer was yet to be observed, he would waker her up to pray the Witr.
Another narration by Muslim reads that when the Prophet prayed the Witr, he would say to her, "O ‘Aa’ishah, get up and pray the Witr."
Moreover, Abu Bakrah, may Allah be pleased with him, narrated, "I went out with the Prophet to pray the Fajr prayer. Whenever he passed by a sleeping man, he would call him for prayer or nudge him with his foot." [Abu Daawood narrated it with a weak chain of narrators, but he did not consider it as weak.]
Allah knows best.