Is it permissible for my uncle, who is mentally ill and works as a caretaker in a mosque but does not clean it, to delegate someone else to do his work in exchange for a portion of his salary while retaining his position? And is it permissible for my grandmother (my uncle's mother) to take his salary?

According to the majority of scholars, a lease (ijarah (definition)) is invalidated by extenuating circumstances, and the lessee (musta'jir) has the option to annul the contract. Therefore, your uncle must inform the official authorities responsible that he is unable to personally perform the mosque service. He is not permitted to appoint someone else in his place without their knowledge, as the contract was made with him. Your uncle can retire due to his illness and continue to receive his salary. If this is insufficient, he can request an increase from the authorities responsible for assisting the elderly and those with chronic illnesses. As for your grandmother receiving your uncle's salary, it is permissible if he gives her permission or if she is a guardian of his wealth. Otherwise, it is not permissible unless she is in need, in which case she may take from it an amount sufficient for her needs, in a reasonable manner.