Mr Johnson also told a Downing Street press briefing on Monday evening that vaccination for those aged 23 and 24 will open from Tuesday 15 June.
He added that the government was bringing forward its target to offer every adult in England a first dose of the vaccine to 19 July – it had originally intended to achieve that by the end of July.
Rising numbers of cases as a result of the Delta variant, which was first identified in India, have forced the government to delay its plans to relax all restrictions for four weeks.
Data published by Public Health England earlier on Monday showed that the vaccines currently in use in the UK were 'highly effective' against hospitalisation with the Delta variant after two doses.
Mr Johnson said: 'By Monday 19 July, we will aim to have double jabbed around two thirds of the adult population, including everyone over 50, or the vulnerable, all frontline health and care workers, and everyone over 40 who received their first dose by mid May.
'And to do this we will now accelerate the second jabs for those over 40, just as we did for the vulnerable groups, so they get the maximum protection as fast as possible.'
He said that if the government proceeded with its original plan to lift all restrictions on 21 June 'thousands more deaths would ensue, that could otherwise have been avoided'.
'By being cautious now we have the chance in the next four weeks to save many thousands of lives by vaccinating millions more people,' the prime minister added.
The Scottish government announced at the weekend that second jabs for over-40s would be brought forward from 12 to 8 weeks.
Data from Public Health England found that after two doses the Pfizer vaccine was 96% effective against hospitalisation with the Delta variant of COVID-19 and the Oxford/AstraZeneca jab was 92% effective. After one dose the Pfizer vaccine was 94% effective against hospitalisation and the Oxford/AstraZeneca vaccine was 71% effective.
England's CMO Professor Chris Whitty told the press briefing that while hospitalisations remained low there had been a 61% increase in the North West since last week and a 50% increase across the country as a whole. 'We expect that will probably accelerate,' he said.
Professor Whitty added: 'The link between people getting an infection and being hospitalised has been substantially weakened [as a result of vaccination], and a much smaller proportion of those infected are going into hospital, but it has not been completely stopped.'
He added that while he didn't think there was an immediate risk of the NHS being overwhelmed it was 'likely' if cases continued to rise on an 'exponential path'.
Mr Johnson said he was 'confident that we will not need more than four weeks' until all restrictions could be lifted.
Last week NHS England announced £20m in funding to help local COVID-19 vaccination sites deliver second doses of vaccine for over 50s and vulnerable groups faster and told CCGs to 'minimise' demands placed on primary care through locally commissioned services not related to the pandemic.