No 10 is increasingly worried that hospitalisations and deaths among double-vaccinated people could rise due to waning immunity as an estimated 4.5 million people have failed to get their booster shots despite being eligible.

Downing Street sources told the Guardian that the gap between those eligible and those jabbed was too wide, ranking it as their major concern ahead of the winter months.

They acknowledged more needed to be done to encourage take-up as waning immunity meant more double jabbed people could get sick and were at greater risk of dying without the extra protection of the boosters.

The UK reported 293 deaths with Covid on Tuesday – the highest level since February – although the figure included all NHS England deaths from Monday owing to reporting delays. There were 33,865 daily reported cases, a drop from more than 40,000 in recent days, and 9,538 people were in hospital with Covid on Monday compared with 6,467 a month ago.

A record 1.6m boosters were delivered last week, with 7m in England since the programme started and hundreds of thousands of people booked in to receive jabs, according to NHS England. However, there are 11.5 million people eligible for their boosters, leaving a gap of 4.5 million.

John Roberts, from the Institute and Faculty of Actuaries, told the BBC on Tuesday expressed worries that not all those in priority groups will have had their third jabs before Christmas.

Multiple studies suggest waning immunity from vaccines. Data from Israel, where 78% of people aged 12 and older are fully vaccinated, showed a large rise in infection rates over the summer. The UK-based Zoe Covid Study app, where users can log their jabs, test results and symptoms, has suggested protection against infection after two shots of Pfizer/BioNTech decreased from 88% at one month to 74% at five to six months, while protection against infection after two Oxford/AstraZeneca jabs fell from 77% to 67% at four to five months.

Prof Tim Spector, the lead scientist on the study, said in August: “In my opinion, a reasonable worst-case scenario could see protection below 50% for the elderly and healthcare workers by winter.” A Danish study suggested the under-65s had about 80% protection for at least six months, while the over-65s had only 47% protection.

The government is pinning its hopes of avoiding further restrictions over the winter on people taking up booster jabs, which are currently offered to over-50s and those with clinical vulnerabilities who had their second jab at least six months ago. Hundreds of walk-in clinics have opened up as part of an NHS campaign.

Boris Johnson spoke about the need for people to realise the risks of not having their boosters over the weekend, saying it would be “tragic” if vulnerable people got seriously ill because they were “overconfident about their level of immunity and didn’t get their booster when they needed it”. “I think people don’t quite realise that the first two jabs do start to wane,” he said.

Roberts, from the Institute and Faculty of Actuaries, said that he was concerned that the first priority groups will not all have received their Covid booster jabs by Christmas, warning that this could start to filter through into hospitalisation and death numbers in the next few days. “My worry is that, at the current rate, we’re still not going to complete the first priority groups one to nine until mid-January,” he told BBC Radio 4.

“And that’s going to mean that there’s going to be a lot of mixing over Christmas with festive behaviour before and then on the day of people who haven’t got that extra protection from the booster, which really does make a huge difference. The analysis that we’re getting from Israel is that it will probably reduce deaths by 80 or 90% even. So it’s important that we get on with it as quickly as possible.”

An NHS spokesperson said: “It is vital people come forward as quickly as possible – there are appointments available across the country and people can book in through the national booking service or use one of the hundreds of walk-in vaccination sites.”