1) My mother is more than 90 and is an alzheimer patient. She remembers to offer prayers but doesn't deliver properly. At times asks someone to listen if she is wrong. Her caretaker is Christian. I live in another city.

2) Mostly she doesn't wakeup for morning prayers as she is restless throughout night. How can we cover up all her namaz?

1) In the above situation there is no need to check her pronunciation (Makhraj ect…) she can pray the way she feels comfortable and not need to correct her.

2) None can perform others namaz as long as he/she is alive ( even if h/she is in the state of unconsciousness) therefore if she is has to perform her Qaza Namaz by any means. But if you look Namaz rules you would find there are many exceptions discussed. So if a person can’t stand and pray, he can sit but if he can’t sit he can perform with making signs with his hands.
But if a person has lost his conscience and doesn’t recover and left this world then nothing will be remained for him as a due.