The prime minister made the pledge at a Downing Street press conference in which he said that hundreds of thousands of jabs would be administered each day by the middle of the month.
Mr Johnson confirmed that almost 1.5 million people in the UK have now received their first dose of the coronavirus vaccine as part of the NHS’ mass immunisation scheme.
He said: “We’ve now vaccinated 1.26 million people in England, 113,000 in Scotland, 49,000 in Wales and 46,000 in Northern Ireland.
“So, all together, nearly 1.5 million people across the UK have now received their first dose and within two to three weeks all of them will have a very considerable degree of immunity.”
The vaccination drive will be helped by the army, which will use “battle preparation techniques” to ensure the speed of the operation, according to the prime minister.
Health secretary Matt Hancock will give the “exact” details of how the government intends to reach its vaccination target on Monday, he added.
Mr Johnson encouraged everyone to do their bit to speed up the process, saying: “You can make a huge difference if you reach all those (priority) groups but we do depend on people coming forward.”
“I really do urge people, when you get notification, when you get a message from the NHS, from your doctor, saying you’re eligible for a vaccine, please take it,” he added.
Mr Johnson’s comments come three days after Brian Pinker, 82, became the first person to receive the Oxford/AstraZeneca vaccine, after it gained regulatory approval in the UK at the end of December.
Although health secretary Matt Hancock described the use of an additional vaccine to the Pfizer/BioNTech one as a “pivotal moment” in the fight against Covid-19, the positive news was tempered hours later by Boris Johnson’s lockdown announcement, which emphasised the scale of the challenge ahead.
For two successive days on Wednesday and Thursday, the UK has recorded more than 1,000 daily coronavirus deaths, a fatality rate not seen since April.