The jabs have also prevented more than 52,600 hospital admissions, data suggests.
The modelling data by PHE and the University of Cambridge cover the period up to July 23.
Previous estimates for the period up to July 9, had suggested about 37,000 deaths and 11 million infections had been prevented.
A further 31,117 positive tests were reported by the UK government on Thursday, the second day in a row in which cases have increased.
It follows a week of steadily declining numbers, with cases halving in eight days.
And 85 people died within 28 days of testing positive for Covid-19, the government reported.
As of Wednesday, 6,034 Covid patients are in hospital, which 853 on ventilation.
About 88% of adults in the UK have had at least one vaccine dose, and about 71% are now fully vaccinated.
England's deputy chief medical officer Jonathan Van-Tam said he hopes the worst of the pandemic is behind us but warned there could be “one or two bumpy periods” ahead.
He disputed a claim made to the Daily Mail by an anonymous senior minister earlier this week that Covid is “all over bar the shouting” as cases dropped.
Prof Van-Tam told BBC News: “I wish it were so. This is not all over bar the shouting. I hope the worst is behind us but I think it’s quite possible that we’re going to have one or two bumpy periods in the autumn and in the winter, not only through Covid, but also through flu and other respiratory viruses as well."
He urged people to get their flu vaccines ahead of the winter, explaining that lockdowns last winter had “completely shut out pretty much all of the other respiratory viruses”.
He said: “If we don’t have further lockdowns, and I hope we don’t, then I think other respiratory viruses like flu are also going to come back this winter, and it’s going to be equally important that people who are called for their flu vaccines actually come forwards and have them this winter.”
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Dr Jamie Lopez Bernal, consultant epidemiologist at PHE, said the latest modelling on the rollout shows the “remarkable impact” vaccines have had.
He said: “As cases have increased, the true scale of protection from the vaccine programme has become clear. Everyone that has come forward for their vaccine has played a part in this vital effort.
“It remains vital that everyone gets two doses of the vaccine, to protect you and those around you from Covid-19. You must book your second jab when invited, to gain maximum protection.”