No Plans As Far As We Know
At the moment, Prince Charles has not expressed any intention to abdicate once it is his turn to be king. It looks like the Queen is also preparing him for his future duties as the monarch by giving him certain responsibilities. For one thing, he has previously placed the Remembrance Day wreath instead of her.
Operation London Bridge
It is hard to imagine having any other monarch but the Queen. However, there is no need to worry as there is a detailed plan for this eventuality. There is allegedly a plan called Operation London Bridge. News of her eventual death will spread among civil servants through this code: “London Bride is down.”