It was announced last week that the head of state has a “form of cancer” and at the weekend he issued a statement praising “all those organisations which support cancer patients”.
Charles has postponed all public-facing duties, but is continuing with behind-the-scenes work on his red boxes of state papers.
The Queen has been carrying out her royal diary of engagements and last week said her husband was doing “extremely well under the circumstances”.
She said the King, who underwent his first bout of cancer treatment last week, had been “very touched” by all the messages of support he has received from the public.
The King and Queen arrived as downpours swept across London and Charles was photographed waving to well-wishers.
Charles's diagnosis followed three nights in a private London hospital where he was treated for an enlarged prostate.
The Prince of Wales may step in to represent his father at events and it has been reported William and his family are spending half-term at their Norfolk home Anmer Hall, which is on the King's Sandringham estate.
The King’s youngest son, Prince Harry, flew from his home in the US to visit him after his diagnosis was revealed.
The Duke of Sussex was reported to have spent about 30 minutes with his father during the whirlwind trip, which he undertook without wife Meghan or children Archie and Lilibet.
Meanwhile, Harry and Meghan on Monday night launched a new website using their Sussex title in a move which could prompt a row with the Royal Family.
Their rebrand from the Archewell website, Sussex.com, appears to emphasise their royal status, with a royal coat of arms, and the message: “The office of Prince Harry and Meghan, the Duke and Duchess of Sussex.”
A picture of the couple smiling at last year’s Invictus Games also greets visitors to the site.
A source told the Daily Mail the use of the royal crest and Sussex titles could spark complaints from the Royal Family.
However, a source close to Harry and Meghan said: “Prince Harry and Meghan are the Duke and Duchess of Sussex. That is a fact. It is their surname and family name.”