On 2 April, the day he emerged from quarantine after testing positive for Covid-19, Matt Hancock stood at the Downing Street podium for the daily coronavirus press conference and made an announcement that was greeted in some quarters by a sharp intake of breath.

“I’m now setting the goal of 100,000 tests per day by the end of this month. That is the goal and I’m determined we’ll get there,” said the health secretary, who had tested positive around a week earlier.

It amazed many observers, but it wasn’t the most ambitious testing target the government had put forward. On 25 March, under pressure on testing and claiming the UK was doing better than other countries, the prime minister said: “We are going up from 5,000 to 10,000 tests per day, to 25,000, hopefully very soon up to 250,000 per day.”

But even the lower numbers have never stacked up. With less than a week to go until his self-imposed deadline, and in spite of buoyant talk by ministers and advisers, many believe that the 100,000 target to which Hancock so firmly committed himself will also be missed.

A massive effort to set up big new laboratories and engage smaller ones, the “little ships”, to help, along with a plan for 50 drive-through centres, has increased capacity. But with a week to go, the UK finds itself with capacity for 40,000 tests a day – and only half that number being carried out. It is well short of the pledge and, more importantly, if the intention is for the UK to test its way out of lockdown, it is not enough.

If the target is missed, there is a risk that blame may be deflected on to the tens of thousands of health workers and key workers themselves. Hancock has already said that they are not turning up at the drive-through centres in the numbers expected. But in fact nurses and doctors have ended up self-isolating because of the long distances they would have to drive to get tested – and there have been stories of some being turned away when they get there.

And as strategists and politicians start to map out the exit from lockdown, the question of resources for local public health teams that have been stripped back over a decade of austerity will become increasingly urgent. The regular – perhaps weekly – testing of those on the frontline and tracking down contacts has to be done locally. It cannot be run, as so much of the epidemic response has been, from Westminster.

But to understand how that pledge has begun to seem so out of reach, it is essential to look at how the government’s approach to it began: slowly. The UK started ramping up testing too late, having abandoned it as a core principle on 12 March, when Johnson declared the virus could no longer be contained. People with symptoms should just stay at home, he said. Only hospital patients would be tested. Thus Britain went to the back of the queue for testing kits, reagents and other commodities that the world was scrambling for.

Testing, followed by contact tracing and quarantine, is the standard public health response to an infectious disease epidemic, supported loudly by the World Health Organization, whose director general said it should be “the backbone of the response in every country”. The UK had other ideas, but as the numbers admitted to hospital rose and NHS staff began to stay at home because they or their family fell ill, the calls for testing became more vocal.

Promises began to be made. In an odd disconnect, NHS and Public Health England had said they intended to reach 10,000 tests a day on 11 March, the day before Johnson declared the containment phase over and said people with symptoms should stay home untested. Just 1,215 people had been tested by that stage.

The issue got hotter. On 18 March, Johnson declared the UK would do 25,000 tests a day, but did not say when. On 29 March, Hancock tweeted that they had reached the 10,000 target and were on course for 25,000. The watchdog organisation Full Fact, however, later said the data did not support the claim. The highest number tested in a day in England at that time had been 9,114.

Facebook Twitter Pinterest Military personnel assist at Haydock Park drive-through testing facility. Photograph: Uk Ministry Of Defence/Reuters

On 2 April, Hancock made the pledge that threatens to “come back and bite him”, according to the Telegraph this week, citing “a No 10 insider”. They would do 25,000 tests a day by mid- to late April, and 100,000 tests a day by the end of the month.

With the prime minister in isolation in Downing Street with symptoms, and a need to quash difficult questions about why the UK lagged behind, the health secretary made some admissions.

Unlike Germany, the UK did not have a huge diagnostics industry, he said, so had been “catching up”. He acknowledged that global demand had led to a shortage of both swabs and reagents. The swabs issue had been fixed but “we are still tackling the reagents issue, which is a global challenge”, he said.

In a rare moment, he admitted that the government had not got everything right: “There will be criticisms made, and some of them will be justified.” But he also had a reassuringly substantial-sounding plan for what was next, from bringing in commercial partners to scaling up Britain’s diagnostic capacity: the “five pillars”.

However many pillars there were, 100,000 tests a day was such a big number that it was assumed the government was including antibody tests to demonstrate people had been infected in the past, as well as the PCR swab tests for people with symptoms.

The government had been enthusiastic about the possibility of a fingerprick antibody test that anyone could do at home, like a pregnancy test. Johnson on 19 March said: “To give you an idea of what is coming down the track, we’re in negotiations today to buy a so-called antibody test, as simple as a pregnancy test, which can tell whether you have had the disease. It’s early days, but if it works as its proponents claim, then we will buy literally hundreds of thousands of these kits as soon as practicable. Because obviously it has the potential to be a total game-changer.”

Public Health England’s director of infections, Prof Sharon Peacock, elaborated on 25 March, telling the science and technology committee of MPs that a simple home test would be available within days.

But it wasn’t to be. Oxford University scientists have been evaluating some of the 3.5m antibody tests Hancock bought from a number of companies. He had placed orders for a further 17.5m. The best of them are 70% accurate. Most are only right half the time. Having previously claimed that it had struck deals which meant no payment would be made until the tests were proven, the government is now trying to get its money back.

So the 100,000 tests are PCR swab tests for current viral infection only, and all the stops have been pulled out to get there. On Wednesday, the Department for Health and Social Care hailed the “biggest network of diagnostic labs in British history”, built in five weeks flat. Three mega-labs would process samples from 27 of the planned 50 drive-through centres. Dozens of universities, research institutes and companies were lending equipment. The UK’s two biggest drug companies, Astra Zeneca and GlaxoSmithKline, are collaborating on a fourth lab in Cambridge.

And yet, only half the daily 40,000 test capacity is being used. Hancock talked of lower than expected demand and has opened the scheme to other key workers outside the NHS, including social care workers, police, prison officers, social workers and benefits staff.

But there is a fundamental problem. Key workers don’t want a long drive at the end of a gruelling shift to get tested.

“What I’m hearing from the frontline is that nurses are driving up to two hours, feeling very unwell with possible symptoms of coronavirus, to testing stations, and sometimes if you haven’t got an appointment you’re turned away and told to come back another time,” Dame Donna Kinnair, general secretary of the Royal College of Nursing, told the health and social care select committee.

The British Medical association estimated on Wednesday that nearly 100,000 health workers were self-isolating because the testing sites were not easy to get to.

Prof John Ashton, former regional director of public health for north-west England, said the government’s battle plan against Covid-19 had been so centralised and top-down that they hadn’t asked people what would work for them in their locality.

“You have to go to them. They should be testing them as they leave work,” he said. “It’s the same issue with immunisation.” Offering nurses flu jabs in the hospital as they arrive or go home has successfully raised rates in the past.

Importantly, testing is not an end in itself. When the UK attempts to lift the lockdown, testing, tracing the contacts of anyone who is positive and ensuring they isolate themselves is essential. Key workers running higher risks may have to be tested weekly. That needs to be run locally, by people who understand the community they live and work in.

But regional public health teams have been decimated. Local authorities were given responsibility for public health in the Andrew Lansley reforms. Cash-strapped town halls have raided the public health budgets, which are down by at least 30%. Public health teams “are a shadow of what they were”, said Ashton.

In the 1970s, Liverpool had 6,000 public health staff, including community nurses, health visitors, food hygiene staff and many others. “Now they’d be lucky to have 30 or 40 staff,” he said. “Bolton has got four in its public health team.”

The epidemic will not behave in the same way in every part of Britain. Once out of UK-wide lockdown, cases will flare in one region or another, and the best firefighters will be local public health teams. These are people who know their communities. It is the local teams, says Ashton, who could have made the next chapter less disastrous than the current one.

“They are the ones that should be leading on contact tracing and testing programmes,” he said. “This can’t be done from Whitehall.” The problem is, in many cases, they barely exist.