Health officials say they are "frustrated" by a lack of progress in expanding UK coronavirus testing.
Prof Paul Cosford of Public Health England (PHE) said "everybody involved" was unhappy testing had not "got to the position yet that we need to get to".
Prime Minister Boris Johnson responded to criticism of the UK's strategy by hailing screening as a solution to "unlock the puzzle" of coronavirus.
It came as a further 569 patients died with coronavirus in UK hospitals.
As of 17:00 BST on Wednesday, the overall number of deaths with the virus in the UK was 2,921.
Mr Johnson, who continues to suffer mild symptoms during self-isolation in Downing Street after contracting the virus himself, has faced widespread criticism over his government's testing strategy.
Around 13,000 tests are available each day against a target of 25,000.
Prof Cosford said testing would hit 15,000 per day "imminently" and that PHE had played its part in ensuring tests were "available to support clinical treatment of patients who need it".
The government has confirmed that 2,800 NHS frontline staff out of half a million in England have been tested for coronavirus since the outbreak began.
Prof Cosford, PHE's emeritus medical director, told BBC Radio 4's Today programme the figure was "nowhere near where we need to get to but it's a good start".
Meanwhile, Labour MP Dawn Butler said her uncle died after contracting coronavirus in hospital following a fall.
"Government needs to test everyone who works in the hospitals without delay," she wrote on Twitter.
And Labour peer Shami Chakrabarti said the slow pace of testing showed "a lack of clarity of what the plan is and how it is going to be executed".
Sir Paul Nurse, chief executive of the Francis Crick research institute - which will soon be able to conduct 500 Covid-19 tests a day - said a Dunkirk-style effort was needed to co-ordinate smaller laboratories and increase test numbers.
"We are a lot of little boats and the little boats can be effective," he said, referring to the evacuation of Allied troops from the beaches of the French city during World War Two.
He added: "The government has put some big boats, destroyers in place. That's a bit more cumbersome to get working and we wish them all the luck to do that, but we little boats can contribute as well."
The daily figures for the number of new cases, deaths and tests virtually mirror those released yesterday.
At this stage, when we are on an upwards trajectory, the fact that there has not been significant increases in terms of cases and deaths can perhaps be interpreted as a good sign.
But what is more important is the trend over a few days - not 24 hours in isolation.
Any improvement will be seen in the number of new cases first.
They were hovering around the 2,500 to 3,000 mark in the five days up to yesterday.
Then the number of new cases jumped to over 4,300, which has been nearly matched today.
There will be hope that this is where it plateaus, before dropping down.
In a video message on Twitter on Wednesday evening, Mr Johnson said that increased screening would be how the UK defeats the coronavirus.
"I want to say a special word about testing, because it is so important, and as I have said for weeks and weeks, this is the way through," he said.
"This is how we will unlock the coronavirus puzzle. This is how we will defeat it in the end."
Mr Johnson said more coronavirus testing would enable staff who were self-isolating - either because they had symptoms or shared a household with someone who was sick - to know if they were safe to work.
Antibody tests - which look for signs of immunity in the blood - could also show who had already had coronavirus and was therefore not at risk of being infected or passing the infection on to others, the prime minister said.
However, these tests are not yet ready for use and it is not clear when they will be.
The prime minister's official spokesman said more than 2,800 people had been tested at drive-through sites and a "significant number" additionally in NHS and Public Health England laboratories, with an update expected later.
The spokesman confirmed that two laboratories, in addition to one in Milton Keynes, were being set up in Cheshire and Glasgow to process tests in the North of England and Scotland.
Health Secretary Matt Hancock has held talks with industry figures, issuing what his department said was a "rallying call" to improve diagnostic capability.
However, some NHS trusts have said they are limited in the number of tests they can carry out due to continued shortages of swabs, reagents and testing kits.
Despite there being capacity for 12,799 daily tests in England, there were only 10,657 carried out on Wednesday.
As of 09:00 on 2 April, 163,194 people in the UK had been tested for the virus, of which 33,718 were confirmed positive.
Meanwhile, Addenbrooke's hospital in Cambridge has become the first in the UK to use a new, much quicker Covid-19 test for staff and patients.
Called Samba Two, it gives a result in just 90 minutes - as opposed to the 24 hours tests currently take - and has been adapted from an HIV test by a small Cambridge technology company.
Businessman and philanthropist Sir Chris Hohn, who is helping make the test more widely available, told the BBC it could be a "game-changer" in helping hospitals cope with the crisis.
In other developments:
- Comedian Eddie Large has died with coronavirus aged 78, his family said
- Loan and credit card payments could be frozen to support those struggling as a result of the coronavirus pandemic
- British Airways is expected to announce it will suspend around 36,000 staff due to the crisis
- A third prisoner in the UK has died after contracting coronavirus at HMP Littlehey in Cambridgeshire. So far, 73 inmates across 27 prisons have tested positive for the virus, as have 15 staff in nine prisons, and four working in prisoner escort and custody services, the Ministry of Justice said
- Up to 3,000 armed forces reservists have been called up to aid the military response to the coronavirus pandemic
- The Treasury is to announce changes to its emergency loans scheme for businesses by the end of the week - after thousands of struggling companies had their applications turned down
- The chair of the digital, culture, media and sport committee has suggested football clubs that refuse to cut salaries of players while cutting the pay of non-players should be hit with a windfall tax. Some Premier League clubs have furloughed non-playing staff.
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