The antibody tests - also known as a serology test - show who has been infected, although it is not yet clear whether the presence of antibodies to the new coronavirus, SARS-CoV-2, confers permanent immunity.
The FDA changed its policy May 4, giving test makers with active products 10 days to submit applications for review.
These fingerprick blood tests - with results coming back the same day - could enable tens of thousands to return to work confident they have built up some immunity.
Sample testing for antibodies across the country indicated that 17% of people in London might have had coronavirus, as against "about 5% or higher" in the rest of the country, Hancock announced at the daily Downing Street press conference.
The test does not need to be sent off to a lab and will be rolled out if it is shown to be effective, Mr Hancock said.
They will know if they have had COVID-19 and can go to work without fear of catching it or passing it on.
They will learn more about who is most at risk from coronavirus, who is more likely to die and who is more likely to transmit it to others.
Hancock said: "This is an important milestone and it represents further progress in our national testing programme".
"As the Government acknowledges, of course, there are still unknowns". The other searches for antibodies, which indicate whether a person was already exposed and fought the virus off.
He said that over 3 million people had been tested for the virus, of which 250,908 tested positive.
"From next week we will begin rolling these out in a phased way, at first to health and care staff, patients and residents" of care homes, Health Secretary Matt Hancock said.
These could include key workers - teachers, train drivers or prison staff. The tests, produced by Roche and Abbott, will be produced over the next several months, Reuters reported. "We hope this testing will help us learn more about how the disease spreads in our community and who it has affected".
The test will be rolled out to a number of A&E departments without access to laboratories, GP coronavirus testing hubs and care homes across Hampshire.
If the pilot is successful the test, developed by United Kingdom manufacturer Optigene, will be rolled out more widely, the department said.
Mr Hancock said the testing scheme was the "guiding star" to "restore freedom in this country" as he announced the new plans to try to track the contagion. "This will enable health and care workers to carry on with their shift or immediately isolate on the same day".