The UK has recorded a massive rise in the number of people testing positive for coronavirus, amid concerns the government has lost control of the epidemic just as people are returning to work and universities prepare to reopen.

Labour has demanded the health secretary, Matt Hancock, give an urgent statement to the House of Commons to explain the increase and why some people are still being told to drive hundreds of miles to have a test.

On Sunday almost 3,000 people in the UK tested positive for Covid-19, a 50% increase in a single day and the highest daily total since May.

“They’ve lost control of the virus,” said Prof Gabriel Scally, a former NHS regional director of public health for the south-west. “It’s no longer small outbreaks they can stamp on. It’s become endemic in our poorest communities and this is the result. It’s extraordinarily worrying when schools are opening and universities are going to be going back.”

As seen in other countries opening up after lockdown, the majority of new cases appear to be in younger people who typically have milder infections than the over-50s. The number of people needing hospital treatment has remained steady, but these lag behind new cases by about two weeks.

Public Health England reported 2,576 new cases on Sunday and 2,988 for the UK overall. “It’s a massive jump,” said Christina Pagel, a professor of operational research at University College London. “There is no way you can look at these figures and feel confident that things are going in the right direction.”

The rise came amid concerns that testing centres were struggling to cope with demand. Many people who sought tests in recent days were advised to take round trips of more than 100 miles because their local centres did not have capacity.

Andy Burnham, the mayor of Greater Manchester, said many people had contacted him in recent days saying they had been offered tests in Leicester and Scotland.

“The fact people are being directed to bizarre locations is yet another example of national test and trace not working. That’s why it needs to be under local control. The danger is someone who is symptomatic in Greater Manchester, where many areas are still classed as high risk, tries to book a test, gets directed to Leicester and thinks ‘sod that’ and then potentially passes on the virus. It is so obvious that the system should always offer you a test at your local centre, it should keep you within your geography.”

A government source said there was significant concern that the UK was “six weeks behind France”, where the trajectory showed more young people being infected, leading to increased hospitalisations of vulnerable groups.

Hancock said the rise was “concerning” but said workplaces should still be operating safely.

“The cases are predominantly among younger people but we have seen in other countries across the world and in Europe this sort of rise in the cases among younger people lead to a rise across the population as a whole, so it is so important that people don’t allow this illness to infect their grandparents and to lead to the sort of problems that we saw earlier in the year,” he told Sky News.

Paul Hunter, a professor of medicine at the University of East Anglia, said while some of the new cases may be because of catch-up from delayed tests over the past few days, it was still “a marked increase”. He said reports of people making long journeys to get tested did not bode well for the autumn and winter when cases are expected to rise. Having people driving around the country with coronavirus would help spread the disease, while focusing testing on hotspots risked missing fresh outbreaks that could be brought under control, he said.

“It’s got to be a better managed and better put together system than the one we have now. If we’re not coping now, it’s going to be awful in two months’ time when case numbers have doubled or quadrupled,” he said.

Jonathan Stoye, a virologist at the Francis Crick Institute in London, said his son travelled 80 miles from St Albans to Gatwick to get a test. “It’s ridiculous. If you want to get people back to work, you’ve got to get the testing system to work, or people won’t go if they are being responsible.”

It took 59-year-old Jackie Cawkwell, who works as an administrator in Nottingham, three days to be offered a coronavirus test close to home. She started feeling unwell on Thursday with Covid symptoms including nausea, diarrhoea and temperature. “When I tried to get a test on the Friday, it only gave me the option of going to Oldham, that’s 57 miles away,” she said. “I tried three times and I was only given Oldham and when you are feeling that poorly, it’s just not feasible to do a 100-plus round trip. I was despairing.”

Labour is likely to ask the Speaker for an urgent question in the House of Commons on Monday to force Hancock to explain the issues. The shadow health secretary, Jon Ashworth, said ministers urgently needed to get a grip on the system’s failings.

Ashworth said the increase in coronavirus cases was “deeply concerning” and a stark reminder that there is no room for complacency in tackling the spread of the virus. “This increase, combined with the ongoing testing fiasco and the poor performance of the contact tracing system, needs an explanation from ministers,” he said.

“Last week Matt Hancock was boasting of his ‘moonshot’ plan to test millions of the population every day but he can’t even get basic testing delivered for people who are experiencing symptoms,” he said.

“What’s more, ministers still aren’t testing care homes staff and residents routinely despite promising to do so. They claim test rationing is to help hotspots but on Friday a Leicester constituent tried to book a local test at a drive-through and was told to travel 55 miles to Sheffield instead. This simply isn’t the ‘world beating’ system we were promised by September. Matt Hancock should come to the Commons today and explain what has gone wrong and how he will fix it.“

A spokesperson for the Department of Health and Social Care said there was a “high demand” for tests, but that capacity was being targeted at outbreak hotspots. They claimed testing capacity would reach 500,000 per day by the end of October, and that new technologies would process tests faster.