Ministers were under pressure on Sunday night to escalate the current lockdown in England amid warnings that current measures may not be tough enough. They came as the health secretary, Matt Hancock, said about 2 million people – including about a third of people over the age of 80 – had already been vaccinated, and the opening of the mass vaccination sites is intended to accelerate the process. More than half a million people over the age of 80 are to receive letters inviting them to attend one of seven large coronavirus vaccination centres opening in England, where they will be able to book an appointment online or over the phone. But, in an interview on Sunday, Hancock also said that, with hospital cases and deaths still rising sharply, the NHS was “probably under the greatest pressure it ever has been”. According to modelling seen by the Guardian, one in five people across England may have had coronavirus, equivalent to 12.4 million people, rising to almost one in two in some areas. It means that across England the true number of people infected to date may be five times as high as the total number of known cases according to the government’s track-and-trace programme. Amid concerns that people are continuing to mix more than they did during the first lockdown, Keir Starmer suggested the current lockdown measures might be inadequate. “They may not be tough enough,” the Labour leader told the BBC’s Andrew Marr Show. Nurseries should probably close, Starmer said. “Quite a lot of people are surprised that primary schools are closed and that nurseries aren’t closed.” Prof Peter Horby, chairman of the New and Emerging Respiratory Virus Threats Advisory Group (Nervtag), said the Kent variant has made the situation “more risky” and that if the infection rate does not slow down then “we’re going to have to be even stricter”. The current lockdown, England’s third, was announced in a televised address by Boris Johnson on 4 January, but Hancock on Sunday, just six days later, refused to rule out further restrictions. Reports suggested that measures could include a ban on exercising with those outside your household, the end of support bubbles, making office workers wear masks and stricter rules for supermarkets such as one-way systems and masks for outside queues. He said he did not want to speculate on what might be needed but that people should follow the rules. To drive home the importance of compliance with the restriction, Prof Chris Whitty, the government’s chief medical adviser, will be doing a round of media appearances on Monday morning to highlight the risks facing the NHS. Whitty, a regular at the No 10 press conferences but not someone who often gives interviews, is expected to stress that the risk from social contact is much greater than it was during the first lockdown because the new variant of the virus is so much more transmissible. Ministers are confident that the vaccination programme will ultimately provide a release from lockdown and mass vaccination centres will open from Monday at the ExCel centre in London, Ashton Gate in Bristol, Epsom racecourse in Surrey, Millennium Point in Birmingham; Robertson House in Stevenage; Etihad tennis club in Manchester; and the Centre for Life in Newcastle upon Tyne. Letters are being sent out to more than 600,000 people over the age of 80 living up to 45 minutes away from one of the sites inviting them to book an appointment. Anyone already offered a vaccination appointment by their local GP can choose which suits them best. Many more GP and hospital vaccination facilities are also due to open this week, as well as the first Boots pharmacy-based pilots, taking the total number of vaccination centres in England to about 1,200. The large vaccination centres, which will also provide the jab to NHS staff, will be the first services to deploy volunteers from St John Ambulance and the NHS volunteer responder scheme who are helping with the programme. The government is committed to ensuring that everyone in the top four vaccination priority groups – care home residents and staff, NHS and care staff, the over-70s and the clinically extremely vulnerable – will have been offered a jab by 15 February. On Monday Hancock will also publish the government’s Covid-19 UK vaccines delivery plan, which he will set out at a No 10 press conference. Hancock said the document was being released “so the public can be assured of the time, effort and resources that have gone and will go into ensuring we protect the British people from the scourge of this virus”. Hancock said the government wanted to vaccinate everyone in the remaining five of the nine priority groups drawn up by the Joint Committee on Vaccination and Immunisation (JCVI) – everyone else over the age of 50, and people over 16 with serious underlying health conditions – by the spring. It wanted to vaccinate all adults by the autumn, he said. Starmer said the priority list should be reviewed after the first four groups have been covered, with perhaps teachers being added as a new category, but the Department of Health said the independent JCVI drew up the list, and a JCVI source said there were no plans to revise the nine groups already included. On Sunday the UK recorded 54,940 further coronavirus cases and 563 further deaths. Although both figures were lower than the numbers for preceding days, the total number of deaths in the last seven days (6,363) is 49% up on the previous week. Ministers are also particularly alarmed about hospital numbers. The most recent UK-wide data showed 32,294 Covid patients in hospital on Thursday last week, with admissions up 35% week on week. That is more than 10,000 higher than the UK total for hospital numbers at the peak of the first wave in April. On Sunday Tracy Nicholls, chief executive of the College of Paramedics, told Sky News that the ambulance service was currently under “unprecedented” pressure and that there were cases of patients having to wait “five, six, seven, eight, even nine hours” to be transferred from an ambulance to a hospital because of a shortage of beds. She also said she had heard of patients with non-life-threatening conditions in areas such as London and the south-east having to wait up to 10 hours for an ambulance to arrive. Hancock said that he expected the government to be able to ease restrictions in the spring but he refused to spell out exactly when he considered spring to be over. One unnamed government official was quoted in the Sunday Times as saying pubs might not be allowed to open in England until the May bank holiday. In a speech on Monday, Starmer will say the government should make putting families first a priority during the lockdown and he will challenge the government to commit to protecting family incomes by preventing the planned council tax rise, reversing the planned cut to universal credit and giving key workers a pay rise. Starmer will also urge people to take inspiration from the Labour government of 1945 when rebuilding after the pandemic, saying: “We can build a country worthy of the sacrifices of the British people, just as we did in 1945 when Attlee’s government built the welfare state from the rubble of war.”