People taking part will have a weekly test which hopes to spot coronavirus cases early on, including people who don't show any symptoms, so that people can self-isolate as soon as possible.
People contacted by the health service's Test and Trace programme in England must self-isolate for 14 days if they have been a recent contact of a confirmed COVID-19 case, but they are not told to test unless they have symptoms.
Britain lacked mass testing capacity when its virus outbreak began and in March abandoned efforts to track and test the contacts of all people with confirmed cases, a decision authorities now say they regret.
But the number of tests being carried out then and now have vastly improved, potentially meaning the rise could be down to more people being tested.More news: Liverpool target Thiago: I have never said I'm leaving Bayern
The British government has announced it is investing in a coronavirus test that gives results in as little as 20 minutes, touting it as the latest milestone in efforts to take the United Kingdom to the front of the global pack in testing for COVID-19.
"Ideally we would be far more advanced in our ability to handle, what we're already beginning to see, an increase in requirement for COVID testing".
Health Secretary Matt Hancock said: "We need to use every new innovation at our disposal to expand the use of testing, and build the mass testing capability that can help suppress the virus and enable more of the things that make life worth living".
Prof McNally told BBC Radio 4: "I don't think the time is right". Of course there are operational challenges from time to time but it works well.More news: Immortals Fenyx Rising: Microsoft Store leaks details
He told Sky News, "At the moment the system works well".
"And we're finding a higher and higher proportion of people in the country who have coronavirus and getting them tests so they can be looked after".
"The pilot showed the at-home saliva sampling kit to be a reliable means of testing for large-scale, regular testing", the Department of Health and Social Care said in a statement.
"We are expanding testing capacity to 500,000 tests a day by the end of October - as well as bringing in new technology to process tests even faster".More news: AJ Styles Confirms He Tested Positive For COVID-19