The Duke of Sussex has enjoyed a brief reunion with the king after taking an overnight flight from California to visit his father following Buckingham Palace’s announcement of the monarch’s cancer diagnosis.

Prince Harry landed at Heathrow shortly after midday following an 11-hour flight from Los Angeles. Travelling alone, the Duchess of Sussex having stayed in the US with the couple’s two children, Harry was pictured being driven from the airport in a black Range Rover with a police escort.

He was driven straight to the king and queen’s London residence, Clarence House, where King Charles was resting after outpatient treatment on Monday for an undisclosed form of cancer.

About 45 minutes later, the king and queen were driven to Buckingham Palace to be flown by helicopter to the king’s private Sandringham estate in Norfolk.

It was King Charles’s first appearance in public since his diagnosis was announced, and he smiled and waved at well-wishers during the drive.

Buckingham Palace announced on Monday that the king, who acceded to the throne 17 months ago, had been diagnosed with cancer after a “separate issue of concern” was noted during treatment for a benign enlarged prostate. It said he did not have prostate cancer.

While no further details have been given by Buckingham Palace about the type of cancer or prognosis, Rishi Sunak appeared to reveal in an interview on BBC Radio 5 Live that the king’s condition had been “caught early”.

When asked on what information his comment was based, No 10 suggested the prime minister had been referring to the fact that Buckingham Palace had noted the swift intervention of his medical team.

King Charles, 75, who has postponed all of his public duties, has already begun his medical care as an outpatient under the supervision of his specialist team of doctors. A palace source told the Telegraph: “He’s on his usual good form in every way, just a little frustrated that his condition has affected not just his own plans but impacted on others.”

Harry last appeared alongside the Windsors at the king’s coronation and left swiftly after the ceremony to fly home for his son Prince Archie’s fourth birthday.

His visit had prompted speculation the duke could also heal his long-running rift with his brother, the Prince of Wales. But Prince William, whose wife, Catherine, Princess of Wales, is recuperating from abdominal surgery, reportedly has no plans to meet Harry.

The two are said to be estranged after Harry’s allegations against his family in an interview with Oprah Winfrey, and, in the months after the death of Queen Elizabeth II, his Netflix documentary and his memoir, Spare.

There were accusations of racism in relation to his son Archie’s skin tone before he was born, with the remarks in the end alleged to have come from two senior royals, and claims that Kensington Palace lied to protect William over reports he allegedly bullied Harry out of the royal family.

Harry also claimed his father was jealous of Meghan and Kate; did not hug him when he told him his mother, Diana, Princess of Wales, had died; and said he believed the king was “never made” for single parenthood but that “to be fair, he tried”.

The duke accused William of physically attacking him and throwing him into a dog bowl in a row over Meghan; teasing him about his panic attacks; and, along with Kate, encouraging him to wear a Nazi uniform at a fancy dress party.

According to Harry, Charles pleaded with his sons to make up during a tense meeting after the Duke of Edinburgh’s funeral: “Please, boys. Don’t make my final years a misery.”

William is returning to official duties this week, beginning with an investiture on Wednesday, having taken time off during Kate’s recuperation.

The heir to the throne is expected to also undertake some duties on his father’s behalf, but counsellors of state, who are appointed temporarily when a monarch cannot fulfil their duties as head of state, are not expected to be needed.

Meanwhile, Anne, the Princess Royal, carried out royal engagements, including an investiture on behalf of the king at Windsor castle on Tuesday, as well as three other engagements.

Well-wishes have been sent to the king from the House of Lords. Opening Tuesday’s session in the upper chamber, the lord speaker, Lord McFall of Alcluith, said: “I know we were all sorry to hear news of his majesty the king’s illness. I am sure the thoughts of the whole house are with the king and his family, and we look forward to his majesty’s full return to public duties in due course.”

The sentiments were endorsed with cries of “hear, hear” across the house.

The palace has called for the king’s privacy to be respected, especially during his treatment, but said he wanted to make his diagnosis public because of his long-running support for cancer charities.

The late queen’s former press secretary Charles Anson told BBC Breakfast that the king felt “very strongly that the condition he has and the treatment for it should be known in general terms” out of a sense of public duty, but that people should not expect “daily bulletins”. The openness about the diagnosis was a “sign of the times”, he said.

He added: “The business of the country and the constitutional role of the monarch continues without really missing a beat,” with other family members able to step in when needed.

The chief executive of Maggie’s, Laura Lee, told BBC Breakfast that as president of the cancer charity, Queen Camilla was “very knowledgeable about the impact of cancer”. She understands that cancer is “a very specific experience for each individual”, but that it also affects family members, who go through a “process of adjustment” after the shock of the news, Lee said.

“The king has done a wonderful job of telling us about his cancer diagnosis, he’s telling others that he’s going through this, they’re not alone, he is with them on that, but he’s also giving his time to his family to come to terms with what’s going on and to see what might lie ahead,” she said.