The coronavirus outbreak has struck at the heart of Britain’s government, with prime minister Boris Johnson and health secretary Matt Hancock testing positive while chief medical officer Chris Whitty self-isolates after experiencing symptoms.

Although all three said they were continuing to work, with only mild symptoms, the dramatic developments throw a question mark over the UK’s leadership during the crisis.

At a Downing Street press conference, senior cabinet minister Michael Gove dodged the question of who would take over if Mr Johnson became incapacitated by a deterioration in his condition. No 10 has previously said that foreign secretary Dominic Raab is “designated survivor” to take the reins of power if Mr Johnson becomes unable to govern.

The UK’s death toll hit 759 on Friday, with the largest daily increase so far of 181, as the virus continued to surge particularly in London.

NHS chief executive Sir Simon Stevens said testing of frontline staff will be rolled out across the health service next week.

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Mr Stevens said numbers of tests should have doubled by this time next week, with GPs, critical care nurses, and other staff in intensive care, emergency departments and ambulance services the first to receive them. He said it was “urgently important” to establish whether frontline staff have the illness to avoid unnecessary isolation.

In a message recorded shortly after he chaired the regular early-morning Covid-19 “war cabinet” by video link today, the prime minister said that he had been suffering from a temperature and a persistent cough and had been advised to take a test by Professor Whitty.

Mr Johnson said that he was self-isolating for seven days but would continue to “lead the national fightback against coronavirus”.

It is understood that his partner Carrie Symonds, who is pregnant, is not self-isolating with him in the flat they share above 11 Downing Street, but is staying elsewhere.

Mr Johnson is believed to be the first major world leader to test positive for the virus.

His diagnosis immediately raised concerns because he has recently been in contact with many of those at the forefront of the UK’s response to the coronavirus outbreak, including Professor Whitty, Mr Hancock and chief scientific adviser Patrick Vallance.

Just hours later, Mr Hancock released his own message to say he too had tested positive, adding: “Thankfully my symptoms are mild and I’m working from home and self-isolating. Vital we follow the advice to protect our NHS and save lives.”

Shortly afterwards, Professor Whitty tweeted: “After experiencing symptoms compatible with Covid-19 last night, in line with the guidance, I will be self-isolating at home for the next seven days.

“I will be continuing to advise the government on the medical response to coronavirus, supported by my deputies.”​

The prime minister was last seen in public on Thursday evening at No 10, where he joined in the national Clap for Our Carers display of support for NHS workers. He was joined by chancellor Rishi Sunak, and the pair observed social distancing advice to keep at least two metres apart. Mr Sunak has not been tested for coronavirus.

Mr Johnson spoke at the regular 10 Downing Street press conference on Wednesday, but Thursday’s conference was led instead by the chancellor, who announced a package of assistance for self-employed workers. Mr Gove appeared behind the lectern on Friday. Downing Street said that Mr Johnson will not be able to take part in the daily conferences while he is self-isolating.

Mr Johnson is known to have been in contact over recent weeks with the Prince of Wales, who has also tested positive for coronavirus. The pair met at a Commonwealth Day service in Westminster Abbey on 9 March. Imperial College scientist Neil Ferguson was also diagnosed with coronavirus days after visiting 10 Downing Street.

His spokesperson said the prime minister had not seen the Queen face-to-face since 11 March, as their weekly audiences were conducted by telephone for the last two weeks.

It is not clear at this point how Mr Johnson contracted the illness, though it is known that other members of Downing Street staff had previously self-isolated because they were showing symptoms.

His official spokesperson said that he noticed symptoms on Thursday afternoon, was tested in 10 Downing Street and received the results at around midnight last night and immediately went into self-isolation.

The spokesperson declined to comment on where Ms Symonds was staying, but said that the medical advice in Mr Johnson’s case was that he should self-isolate for seven days.

The seven-day guidance applies only to people living alone, while those sharing their homes with family members are told to self-isolate for 14 days, to give time to be sure whether others have caught the infection from them.

“The prime minister of course follows all of the guidelines which have been issued by Public Health England in full, and his circumstance is such that he will be required to self-isolate for seven days,” said the spokesperson.

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Daily Telegraph journalist Camilla Tominey, a friend of Carrie’s, said Ms Symonds had moved into the couple’s four-bedroom home in Camberwell, south London, some days ago.

Ms Tominey told ITV’s This Morning: “She’s in Camberwell with Dilyn the dog so she will not have had any contact with the prime minister over the last few days.”

Ms Symonds, whose baby is due in the early summer, is expected to sit out her partner’s self-isolation in Camberwell, where she will face an anxious wait to see whether she has herself caught the virus. Sufferers are believed to be contagious for several days before showing symptoms.

To allow Mr Johnson to continue working, the chancellor’s residence at 11 Downing Street has been turned over in its entirety to serve as the prime minister’s workplace, with Mr Sunak working from his family home. Mr Johnson told colleagues at cabinet on Tuesday that wherever possible they should work from home, and most have been doing so.

Video conferencing facilities have been installed in the downstairs office and study being used by the prime minister in No 11 and doors joining the building with No 10 have been closed off.

All staff working with the prime minister are observing the advice to stay two metres away from him, and meals are being left outside his door for him to collect.

The weekly cabinet meeting took place by video call for the first time ever on Tuesday, with only Mr Johnson, Mr Hancock, Professor Whitty and cabinet secretary Sir Mark Sedwill – the country’s most senior civil servant – attending in person.

Mr Johnson’s spokesperson said stringent efforts were being made to ensure that the prime minister did not pass on the infection to colleagues.

“Here in No 10, we’ve been observing the advice on social distancing,” he said. “You will have seen the images released of cabinet, where we have wherever possible used video-conferencing. You will have seen the prime minister ensuring he was a safe distance from colleagues when taking part in press conferences.

“From the moment he had symptoms, he took steps to make sure he wasn’t in close contact with anyone. He did really want to take part in clapping NHS staff, it was something he felt was important. But in taking part in that national moment, he ensured he didn’t come into any form of close contact with anyone.

“He stood outside No 10 at a very significant distance from the chancellor and he took part in the applause and then immediately after that he placed himself in self-isolation.”

In his video message, Mr Johnson said: “Hi folks. I want to bring you up to speed with something that’s happening today, which is that I have developed mild symptoms of coronavirus – that is to say, a temperature and a persistent cough.

“On the advice of the chief medical officer, I have taken a test that has come out positive, so I am working from home. I am self-isolating and that’s entirely the right thing to do.

“But be in no doubt I can continue, thanks to the wizardry of modern technology, to communicate with all my top team and lead the national fightback against coronavirus.”

Mr Johnson added: “I want to thank everybody involved, above all our amazing NHS staff. It was very moving last night to join in that national clap for the NHS.

“But it is not just the NHS, it is our police, our social care workers, teachers, everybody who works in schools, DWP staff.

Boris Johnson and Rishi Sunak take part in the national Clap for Our Carers event on Thursday evening (PA)

“It is an amazing national effort by public services, but also by every member of the British public who is volunteering – an incredible response, 600,000 people have volunteered to take a part in a great national effort to protect people from the consequences of coronavirus.

“I want to thank you, I want to thank everybody who is working to keep this country going through this epidemic.

“And we will get through it and the way we are going to get through it is, of course, by applying the measures you will have heard so much about.

“The more effectively we all comply with those measures, the faster our country will come through this safely and the faster we will bounce back.

“Thank you to everybody who is doing what I’m doing – working from home to stop the spread of the virus from household to household.