My mom has very bad arthritis in her right hand and she cannot take water for wudhu in right hand (she cannot move her wrist and hand) and she use her left hand to take water. Also for Tayammum too she cannot use her right hand because of severe pain. What is the ruling for her?

Since her right hand is in severe pain and from what I've understood she cannot use that arm for wudhu therefore when doing wudhu of the left hand when she is required to take water in the right and put it on the left, instead she can just hold her hand under the tap and let the water reach everywhere on the hand.

Similarly when it comes to the wiping of the head and feet, she can use the hand that she feels comfortable