The Council of State usually takes charge when the monarch is abroad or temporarily incapacitated, and are given their authority in each specific case by Letters Patent.
The members of the Council were laid down by the 1937 Act, and are the Monarch’s Consort and the first four people in the line of succession who are of age, British subjects and living in Britain. So currently they are DOE, Charles, William, Harry and Andrew.
When Charles becomes King, they would be Camilla, William, Harry, Andrew and Beatrice (unless, of course George is of age by then!)
If William became King before George is of age, the Counsellors would be Kate, Harry, Andrew, Beatrice and Eugenie.
Of course, it’s always possible that Parliament will make different arrangements - when George VI died, the Queen Mother was made a Counsellor, though she was no longer Consort of the reigning monarch nor in the line of succession. Whether Parliament would do the same for Camilla when Charles dies is unknown.