A 19-year-old with no existing medical issues has also died and the UK saw its biggest day-on-day rise in the number of deaths since the outbreak began - up 381 on 24 hours previously to a total of 1,789.
Mass community testing
Critics have warned that mass community testing was the only safe way of lifting the lockdown without risking a fresh outbreak of the virus.
NHS staff have also expressed frustration they are being forced to self-isolate just as they are most needed, because tests are not available to show whether they are clear of the disease.
The Royal College of Physicians, the Royal College of Nursing and the British Medical Association all say testing of frontline staff is desperately needed.
Cabinet minister Michael Gove acknowledged at the daily No 10 news conference on Tuesday that they needed to go "further, faster".
However, he said a shortage of the chemical reagents needed for the tests was proving to be a "critical constraint" on the Government's capacity to ramp up capacity.
8,000 tests each day
Currently about 8,000 tests a day are being carried out, despite ministers having previously claimed to have met a target of 10,000 a day.
A further target of 25,000 tests a day is not now expected to be reached until the end of April. In contrast, Germany is testing about 70,000 a day.
With up to one in four hospital doctors reportedly off sick or self-isolating, Chris Hopson, the chief executive of NHS Providers, expressed concern there was limited capacity to test staff.
He said when limited testing was introduced at the weekend, it suggested many could safely go back to work.
"What was interesting is that of the members of staff who had been self-isolating for 14 days because they had a family member who potentially has coronavirus symptoms, 15% of them were positive," he told BBC2's Newsnight.
“Break the chain of transmission”
"It is internationally proven as the most effective way of breaking the chain of transmission," he said.
"However difficult it is to source the reagents to ramp up the capacity of laboratories up and down the country, it is essential that mass community testing is part of our national strategy."
Mr Gove said Prime Minister Boris Johnson and Mr Hancock, who are both self-isolating after testing positive for Covid-19, were working with companies worldwide to get the materials needed.
The Chemical Industries Association acknowledged demand was "escalating" but said reagents were being manufactured and delivered to the NHS.
"Every business here in the UK and globally is looking at what they can do to help meet the demand as a matter of urgency," it said.
Earlier, it emerged that a 13-year-old boy had died after testing positive for coronavirus. Ismail Mohamed Abdulwahab is thought to be the youngest victim of the disease in England.
His family said they have been left "beyond devastated", adding that to their knowledge he did not have any underlying health conditions.