The number of new cases of Covid-19 being diagnosed is still much too high to allow any easing of the lockdown soon, leading scientists have warned, as the virus death toll in UK hospitals passed 20,000 on Saturday.

The home secretary, Priti Patel, described the figure as a “terrible milestone” and a “deeply tragic and moving moment”. She said it showed the need for the British public to “stay strong” and remain at home for the foreseeable future.

A further 813 deaths were reported in hospitals, taking the UK total to 20,319. This figure does not include deaths from Covid-19 in care homes, hospices and in the community.

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As ministers came under increasing pressure to ease the lockdown from the business community and Tory MPs concerned at the plight of small firms in their own constituencies, scientists said the drop in new coronavirus cases being reported daily was disappointingly slow.

Professor John Edmunds, a member of the government’s Sage group of Covid-19 experts, said if the lockdown was eased now, the newly enhanced testing and contact tracing system being put in place would be swamped.

“The strategy behind plans to lift the lockdown is based on the idea [that] you could then control the epidemic by testing people for infections before tracing their contacts,” Edmunds said.

Facebook Twitter Pinterest Police patrolling Victoria Park, east London on Saturday. Photograph: Justin Tallis/AFP via Getty Images

“However, if we lifted the lockdown now, the testing and tracing system would be overwhelmed. We will have to get case numbers down a lot lower than they are now before we can think of lifting current regulations.”

Professor Keith Neal of Nottingham University agreed that the number of patients being taken to hospital with Covid-19 remained far too high. “This daily figure peaked on 5 April with 5,903 cases. This Saturday it stood at 3,583,” he added. This latter figure was boosted by an extra 1,330 new cases of infected care and health workers, which brought Saturday’s overall total to 4,913.

“It has therefore taken three weeks for numbers of hospitalised Covid-19 patients to decline from a daily total of 5,903 to 3,641.”

Professor Paul Hunter, of the University of East Anglia, added: “There is no doubt this rate of decline is disappointing. Certainly it is far too high to consider lifting lockdown restrictions at present. We need to get numbers down to a few hundred new cases a day before we can do that. Such a decline could take months.”

With no firm indication being given by the government on when social distancing rules might be eased, prime minister Boris Johnson will return to Downing Street on Monday after convalescing at Chequers, having fallen seriously ill with Covid-19 earlier this month.

The prime minister is facing an acute dilemma over the lockdown as members of his cabinet and MPs in his party argue over whether it should be eased to save the economy from collapse – the predominant view in the Treasury – or whether the public should be told to double down on restrictions to reduce the death toll. Matt Hancock, the health secretary, is pushing the more cautious approach.

Former chancellor Philip Hammond said on Saturday it was time that the government announced plans for an exit strategy, while Graham Brady, chairman of the 1922 Committee of Tory backbenchers, said MPs wanted to see a strategy for desperate businesses. He said: “All members of parliament must be receiving representations from businesses large and small needing further assistance, or some sense of when they can start to plan for at least a partial release from these measures.”

But Patel insisted the government’s five tests for easing the lockdown had to be met in full. “It is clear that is not right now,” she said.

Amid signs that public confidence in the government’s handling of the crisis is waning in a new Opinium poll for the Observer, ministers are facing an uphill battle to meet their target of testing 100,000 people a day for Covid-19 by the end of April.

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As of 9am, a total of 640,792 tests had taken place, with 28,760 being conducted on Friday.

Downing Street said the prime minister had said he was “raring to go” after being given the green light by doctors, who had been closely monitoring his progress since he was discharged from St Thomas’ hospital two weeks ago. Sources said Johnson’s return “marked a tightening of grip from the PM”, who held a series of calls and teleconference meetings with key ministers last week.

But on Saturday, Labour leader Keir Starmer turned up the pressure on the PM, writing to Johnson with a warning that the UK risked “falling behind the rest of the world” by refusing to discuss an exit strategy from the lockdown.

Starmer wrote: “The UK government is behind the curve on this. I fear we are falling behind the rest of the world. Simply acting as if this discussion is not happening is not credible … The British public have made great sacrifices to make the lockdown work. They deserve to be part of an adult conversation about what comes next. If we want to take people with us and secure their consent, this is necessary now.”

Meanwhile the Department of Health and Social Care announced that mobile testing units, operated by the armed forces, would travel around the UK to increase access to coronavirus testing.

On Saturday the website offering tests to key workers was overwhelmed for the second day running by 10am, with people in the south of England reporting that the only tests available were in Scotland.

The DHSC said the new units would respond to areas of highest demand, travelling to test frontline workers and the most vulnerable at sites including care homes, police stations and prisons.