The Queen has described King Charles as doing “extremely well under the circumstances” at her first public engagement since his cancer diagnosis was announced.

Camilla said her husband, who underwent his first bout of treatment earlier this week, was “very touched” by all the messages of support he has been receiving from the public.

The king, 75, has been spending his time at Sandringham in Norfolk after his diagnosis, which was announced by Buckingham Palace on Monday evening. The form of cancer has not been disclosed.

The Queen left Charles at Sandringham to fulfil her royal engagement at Salisbury Cathedral. Among the first people she met were medical staff from Wiltshire Air Ambulance on Thursday.

Ben Abbott, 40, a critical care paramedic, said: “I do hope his majesty is doing well ma’am, we’re all really sad to hear the news.”

The queen replied: “Well he’s doing extremely well under the circumstances, he’s very touched by all the letters and the messages the public have been sending from everywhere – that’s very cheering.”

The king has postponed all public-facing duties but is continuing with behind-the-scenes work on his red boxes of state papers.

His son, the Prince of Wales, may step in to represent his father at an event but no engagements are scheduled at the moment.

On Wednesday, Downing Street took the unusual step of confirming that Rishi Sunak would be phoning the king after an agreement with the palace to disclose the information.

The prime minister’s official spokesperson said: “We don’t in general and we are not going to get into the habit of commenting on the PM’s conversations with the king. But we have agreed with the palace in this specific instance to confirm that they will be speaking on the phone later.”

Initially it was speculated that the formal weekly audience between head of state and prime minister, normally held in person and part of the monarch’s core constitutional duties, would be conducted over the phone.

However, a palace source clarified it would be a personal call for the prime minister to wish Charles well and a speedy recovery, rather than an audience.

No audience had been scheduled for the next fortnight, partly in keeping with the pattern set by the late Queen Elizabeth II for this time of year, and also to allow the king to recover from his benign prostate enlargement procedure.