Hundreds of people in line at Lambeth, south London, cheered and applauded as Charles and William emerged on Saturday, before they thanked those waiting to pay their respects to the Queen at Westminster Hall.
They were later joined by the Princess of Wales and the Queen Consort at a Buckingham Palace lunch with governors-general from the Commonwealth realms.
Kate was seen deep in conversation with Camilla.
Earlier, in the line, one woman offered the King her condolences as he shook her hand, and another shouted: “I can’t believe this.”
Dozens shouted “hip hip hooray” as Charles and William moved down the line, stopping for a few moments with each person.
Several people called their friends and relatives on the phone to tell them what they had just witnessed.
Others also shouted “God save the King” and “God save the Prince of Wales” as each passed.
William could be heard discussing how long people had waited and whether they were able to keep warm.
Several people cried after meeting him, and one woman told him: “You’ll be a brilliant king one day.”
Charles left before William, both of them in cars surrounded by police vehicles.
Later, they met the governors-general of Antigua and Barbuda, Australia, the Bahamas, Belize, Canada, Grenada, New Zealand, Papua New Guinea, Saint Lucia, Saint Vincent and the Grenadines, the Solomon Islands and Tuvalu.
Kate, who wore her hair down, was pictured smiling at the Queen Consort, who was holding a drink with a slice of lemon in it, as they chatted.
At one stage the Princess, who was wearing a long three-strand pearl necklace, placed her hand on the arm of Saint Lucia governor-general Cyril Errol Melchiades Charles as they spoke.
Meanwhile, the Earl and Countess of Wessex were met with cheers as they spoke to well-wishers outside Buckingham Palace.
The Queen’s youngest son told the crowd: “I know that my mother would really appreciate this fantastic support.”
Sophie collected flowers and was given paper tributes by some.
The queue for the lying in state began to grow in Southwark Park, with a steady stream continuing to join the end of the line
By mid-afternoon, the current wait time was an estimated 16.5 hours.
The King has also thanked emergency service staff for their work during the mourning period.
Charles met London’s Metropolitan Police Commissioner Sir Mark Rowley and was briefed by Deputy Assistant Commissioner Jane Connors and Commander Karen Findlay, who are leading police operations around London ahead of and during the funeral of the Queen.
Home Secretary Suella Braverman and mayor of London Sadiq Khan were also present.
Early on Saturday, the Department for Digital, Culture, Media and Sport’s queue tracker warned people not to travel to join the back of the queue.
It changed the guidance at 8am, saying the line had reached Southwark Park with wait times “at least 24 hours”.
Although it was a very bright morning in central London, temperatures remained cool, particularly in the shade, at about 12C.
Later, the Queen’s grandchildren will stand guard around her coffin.
William will be joined by his brother the Duke of Sussex, wearing uniform, at Westminster Hall.
Harry, who saw action on the front line during two tours of duty in Afghanistan, has previously been denied the chance to wear military uniform as he publicly mourns, because he is no longer a working royal.
But royal sources say the King has decided his youngest son can wear uniform for the vigil. He will stand at the foot of the coffin, with William at the head.
William will be flanked by his cousins Zara Tindall and Peter Phillips, while Harry will be with Princesses Beatrice and Eugenie, with Lady Louise Windsor and her brother Viscount Severn at the middle of the coffin.
On Friday the Queen’s children – Charles, the Duke of York, the Princess Royal and the Earl of Wessex – held their own vigil.
Final preparations are under way for the funeral on Monday, with politicians and royal dignitaries from around the world expected to arrive throughout the weekend.
Prime Minister Liz Truss concluded meetings with her counterparts from Australia and New Zealand at the Government’s Chevening country residence.
The Prime Minister spoke to Australian PM Anthony Albanese and New Zealand’s PM Jacinda Ardern and is expected to hold a call with the Crown Prince of Saudi Arabia, Mohammed bin Salman, on Saturday.
She also spoke to the UAE president and ruler of Abu Dhabi, Sheikh Mohammed bin Zayed Al Nahyan.
The Prime Minister will have an audience with the King before attending his reception for visiting heads of state at Buckingham Palace on Sunday.