Hospitals sought to take over the operation of a flagship government coronavirus testing centre from the accounting firm Deloitte after severe failings in the service led to the test results of NHS staff being lost or sent to the wrong person, the Guardian can reveal.

The drive-through centre, at Chessington World of Adventures, in Surrey, was among the first in what will be a network of about 50 regional facilities, trumpeted by the health secretary, Matt Hancock, as key to delivering on the government promise of 100,000 tests a day by the end of April.

On Thursday the government insisted it was on track to meet the target, increasing the tests from less than 10,000 at the start of April and just 23,000 tests on 14,000 people on Wednesday.

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Deloitte was hired to help scale up testing nationally, and is understood to be handling logistics across a number of the sites, working alongside other private firms such as Serco and Boots.

Deloitte says it does not run or manage the testing centres, but supports the Department of Health.

In revelations that raise concerns over the rapid outsourcing of testing during the pandemic, chief executives at hospitals in south-west London are understood to have had talks about removing Deloitte and running the Chessington drive-in testing centre themselves.

The alarm was sounded when doctors and nurses complained that test results never arrived, or that they had been sent someone else’s results. In other cases, the test centre, which is about 12 miles south-west of central London, was unable to ring through with the diagnosis because of a failure to record correct phone numbers.

In an email written in early April, and seen by the Guardian, the chief executive of Epsom hospital said: “Deloitte who have been commissioned by the Department of Health directly for this are not running this as well as we would like … [We] are asking whether we can take over the running of the Chessington centre because we really need it to work much better than it is.”

It is understood that the request to remove the firm was made in the first week of April, but that it was set aside after performance improved.

The Department for Health said late Thursday that 49 people had experienced delays, and 39 had now received their results. The ministry said it would continue to investigate the remainder and offer any workers still waiting a new test.

“We are also implementing further processes within the department to ensure rapid investigation of outstanding test results that are not provided to those tested within 48 hours,” a spokesman said.

However, Epsom hospital stopped sending staff to Chessington 10 days ago after finding a more convenient alternative, it said. Staff are now swabbed at work and the samples sent to the laboratory at St George’s hospital, south London for analysis.

Epsom could soon be able to do the analysis itself, having ordered four machines made by a company spun out from Cambridge University, which can complete the process in 90 minutes.

The Chessington centre’s problems appear not to have been entirely resolved. A paramedic for the south-east coast ambulance service, who asked not to be named, said he had been tested along with a group of colleagues on 3 April. They were still waiting to hear from Chessington.

“None of us from my place of work, tested on that Friday, have had our results,” he said. “We were told our samples were taken to a lab in Southampton. It’s three weeks since the test and I don’t think I’m ever going to get the result. No one’s been in touch other than my managers to say they’re chasing it.”

The paramedic questioned the involvement of Deloitte, which specialises in management consultancy, tax and accounting but has little experience in running public services. “I’m not sure what is essentially an accountancy company can bring to this.”

A doctor at Epsom hospital, who was among the first to be tested at the end of March while self-isolating at home with symptoms, said she was still waiting to hear; after making enquiries she was told Deloitte could only release information to the person being tested, and not the organisation referring them. She has since been tested by her hospital.

Asked to comment, Daniel Elkeles, who runs Epsom, and its sister hospital, St Helier, in Carshalton, Surrey, said there were “clearly teething problems at Chessington at the start of this contract”.

“We are now assured that those teething problems are fixed and that the service is reliable. But it’s much more convenient for our staff to be swabbed [at the] Epsom and St Helier sites, and for us to send the swabs to St George’s for testing. This means our staff don’t have to travel to Chessington and also means the facility has more capacity for other parts of the health and care system.”

Normally, government contracts over a certain value are advertised publicly and awarded after a competitive tender.

However, it is understood that the health department appointed Deloitte without competition, under an obscure piece of legislation introduced in 2015 which ministries have been told they can use during the pandemic to avoid lengthy tendering processes.

A Deloitte spokesman said: “Deloitte, alongside many other public and private sector partners, is supporting DHSC (the Department of Health and Social Care) to help accelerate and scale testing capacity for the national Covid-19 testing programme.”