The importance of King Charles shouldn't be underestimated despite the fact that the monarch in the United Kingdom is largely ceremonial, former British Prime Minister David Cameron said in an interview with "CBS Evening News" anchor and managing editor Norah O'Donnell.

The interview comes as King Charles III takes the throne in the United Kingdom after the death of his mother, Queen Elizabeth II. The queen's funeral is scheduled to be held on Sept. 19.

Cameron is one of 15 prime ministers who served during her 70-year reign. Just one week ago, Queen Elizabeth undertook one of her last official acts before her death: Accepting Boris Johnson's resignation and inviting incoming Prime Minister Liz Truss to form a government.

Queen Elizabeth had met weekly with each prime minister in what is called a "private audience." Cameron told O'Donnell that while he was in office from 2010 until 2016, he held practice audiences with Charles to prepare him for when he took the crown.

"He knew his mother couldn't last forever," Cameron said. "Although for those of us, you know, born since 1952, we've always had the queen. And it feels like a rock of our lives has disappeared. So, I think he knew one day he would take on the role. He's thought deeply about it. He wanted to get every part of it right."

O'Donnell asked what kind of real power the new king can have.

"In our system, the king now is the sovereign, but Parliament is sovereign," Cameron responded.

"So it is mostly ceremonial, but we shouldn't underestimate its importance in two ways," Cameron continued. "One is, I think that our system of constitutional monarchy and democracy has brought us great success and stability. The politicians down here can fight it out all they want. We have a unifying figure at the head."

"And the second thing I'd say and I feel this very deeply having watched Queen Elizabeth II in action is she was the world's greatest public servant, and one of the world's greatest diplomats," Cameron added. "Look at what she did to help bring Britain and Germany together after the war. Look at what she did to help with the transition to a non-racial South Africa. Look what she did when I was prime minister. Her visit to the Republic of Ireland and healing so many of the wounds in that relationship was remarkable only she could have done."

When Queen Elizabeth took the throne, she was the head of state in 32 countries that made up the Commonwealth. Now that has been whittled down to 15 countries for which King Charles serves as head of state, and some have already revived debate about leaving the Commonwealth.

Cameron said it would be "their decision" if any of those remaining countries wish to leave. "But I think people can see, and they see this week, that having a constitutional monarchy where your head of state is above politics and a symbol of unity and a symbol of service and duty, and your politicians can fight out the issues underneath that. It's a good system," he continued. "It's served us well, and it's served others well, too."

On a lighter note, Cameron, having had so many private audiences with Queen Elizabeth, recalled that she had a "very good sense of humor" — and was a "phenomenally good driver."

Queen Elizabeth II was the longest-serving British monarch in history when she died last week at 96. On Monday, Britain's two houses of Parliament came together at London's Westminster Hall to express their condolences to King Charles and his wife, Camilla, the queen consort.

"Parliament is the living and breathing instrument of our democracy," the king told the gathered politicians. "While very young, her late majesty pledged herself to serve her country and her people and to maintain the precious principles of constitutional government which lie at the heart of our nation. This vow she kept with unsurpassed devotion. She set an example of selfless duty which, with God's help and your counsels, I am resolved faithfully to follow."

King Charles then flew to Edinburgh to escort his mother's coffin along with his three siblings, Princess Anne and Princes Andrew and Edward.