•Details of the cancer have not been disclosed, but Kensington Palace says it is confident the princess will make a full recovery.
•Meanwhile, the King's nephew, Peter Philips, has told Sky News Australia that Charles is in "good spirits" but "frustrated" that medical treatment was preventing him from returning to royal duties.
People should give the Prince and Princess of Wales "time to heal" after Catherine's cancer diagnosis, a former royal spokesman has said.
Paddy Harverson, who previously worked for the couple, defended Kensington Palace's handling of the announcement.
"This is not just an institution, it's a family... you also have to remember they're human beings," he told the BBC.
Catherine revealed on Friday that she had begun treatment, after weeks of speculation about her condition.
She said cancer was discovered after she underwent abdominal surgery.
Details of the cancer have not been disclosed, but Kensington Palace says it is confident the princess will make a full recovery.
Meanwhile, the King's nephew, Peter Philips, has told Sky News Australia that Charles is in "good spirits" but "frustrated" that medical treatment was preventing him from returning to royal duties.
It comes six weeks after King Charles paused his public appearances following a cancer diagnosis himself.
The two were briefly treated at the London Clinic private hospital at the same time.
"I suppose I'm biased, some of them are friends of mine, but I think it's very difficult when you're in that situation," Mr Harverson told the BBC's Sunday with Laura Kuenssberg programme.
"You have to give them the space so while there might be communications... you also have to remember they are human beings and you have to follow their lead to a certain degree," he added.
The couple have faced intense public speculation about Catherine's health, since her operation in January for a condition that had not been revealed.
Catherine, 42, has not attended any official events since Christmas.
A photograph of her and her children, released on Mothers' Day, was met with a frenzy on social media due to inconsistencies in the picture.
In a statement, Catherine later apologised for "any confusion" caused by the photo, and said that "like many amateur photographers, I do occasionally experiment with editing".
The Sun's editor Victoria Newton told the BBC that Catherine decided two weeks ago to time her announcement about cancer for the last day of her children's school term.
"Her priority was protecting her three children. She didn't want them going to school being asked even more than they already were," Ms Newton explained to Laura Kuenssberg.
The Times, citing a "close friend" of Catherine, reports that the princess decided to record a video message instead of publishing a statement, and that she wrote "every word of it" without input from advisors.
Mr Harverson said he had "no issue" with how the Royal Family had dealt with the announcement of Catherine's diagnosis, and criticised the "permanent death loop" of speculation on social media.
"I am sure she wanted the picture to be as best as possible and she apologised for it so everyone should have moved on," he said. "I really do think we should give them time and space".
Mr Harverson challenged the suggestion that the Royal Family was more fragile because of recent illnesses.
"We just have to come to terms with the new reality, there's fewer of them," he said.
"They will get over this," he added. "I am highly confident that the King, who I know well, is incredibly strong, very resilient, a great spiritual person."
The Sun has called for "social media trolls, idiotic conspiracy theorists and sniping media critics" to "lay off Kate" - days after it published a video of Kate and William at a farm shop in Windsor.
Ms Newton defended her decision to publish that video.
"It was very important to change the narrative," Ms Newton told the BBC. "A lot of the rest of the media then started saying 'Back off Kate'".
She said that before publication she had been in touch with Kensington Palace and was assured "there was no problem with us running those images".
Meanwhile, also appearing on the Sunday with Laura Kuenssberg programme, Imran Ahmed discussed the viral claims made on social media about Catherine in recent weeks, and warned of the dangers of the platforms.
Mr Ahmed, chief executive of the Center for Countering Digital Hate, stressed that social media is "designed to take the most controversial information and... push it to the front", making certain topics and viewpoints appear more popular than they really are.
"Social media is not the vox populi, nor is it a safe and clever place to get information from. It can be deeply distortive as a lens on the world," said Mr Ahmed.
He claimed that social media firms have the means to tackle the spread of harmful conspiracy theories on their platform, "but they choose not to".
Separately, a Kensington Palace spokesman has said the Prince and Princess of Wales were "enormously touched by the kind messages" they have received, and that they were "grateful for the understanding of their request for privacy at this time."
The couple have already said they will not attend this year's Easter Sunday service. It is unclear if King Charles will lead his family to church.
Buckingham Palace say they are "hopeful" he will be able to join the family next weekend.
The King has said he is "so proud" of his "beloved daughter-in-law" and that he and Camilla are in the "closest contact" with her.
The BBC understands that the Duke and Duchess of Sussex have reached out to William and Catherine since her diagnosis.