The monarchy will gradually evolve under King Charles, former prime ministers have suggested.
Gordon Brown said he believes the new sovereign will bring in a more informal, Scandinavian-style monarchy in the years ahead.
“I think that what Prince Charles has already indicated is that the monarchy is going to be smaller,” he told the BBC’s Sunday With Laura Kuenssberg programme.
“It’s going to be more like a Scandinavian monarchy in the future, but not in a bad way – more informal.
“He stopped as he entered Buckingham Palace and talked to people in the crowd, and that was a signal that he was sending that he wanted people to feel that he was approachable.”
Theresa May said while she expects the monarchy to continue to adapt to changing times, the public expect a sense of continuity.
“If you look at the royal family, they have been steadily evolving a different approach, a different way of doing things over time and I am sure King Charles will continue to take that forward,” she told the programme.
“Of course, he is a different person and he may want to change things in some ways, but I think, critically, as the Queen did, any change in the way things are done would be done gradually and very carefully.
“I think they (the public) will expect that sense of continuity, but also that sense that the world has been changing, continues to change, so a willingness to evolve alongside that.
“I think that what they will hope and what they will get is that deep interest in people which Her Majesty had. And King Charles, I believe, has that too – a deep concern for people.”
David Cameron said the new King is well-prepared for what lies ahead, having served “probably the longest apprenticeship in history”.
He said that when he was in office, he had audiences with the then-prince of Wales so Charles could prepare for the day when, as sovereign, he holds weekly meetings with the prime minister.
“From what I saw he will be brilliant at that job; brilliant at listening, brilliant at asking questions, giving wise advice and sage counsel. This has probably been the longest apprenticeship in history,” he told the programme.
Mr Cameron said that, like his mother, the new King is a “superb diplomat” and predicts he will prove a “very worthy successor” when it comes to supporting the British government abroad.
“I saw him in action at Commonwealth heads of government meetings and he knows every leader personally. He interacts with them brilliantly,” Mr Cameron said.
“The soft power that the British monarch brings to help a prime minister and a government with all those international relations, it was obviously outstanding under Queen Elizabeth II.
“I think you will see Charles III will be a very worthy successor in that regard.”
Mr Brown said he believes the King will travel widely.
“I think what he was saying when he spoke on Friday was that he is going to set aside his charities and interests in favour of concentrating 100% on the duties of the monarch,” he said.
“I think that will involve a huge amount of travelling. I think that will involve travelling not just in Britain but across the world.
“That is a good thing for Britain, that we have a monarch who is prepared to be outward looking, who is prepared to speak countries right across the world.”