Video report by ITV News Political Correspondent Romilly Weeks
The prime minister said four million people in the UK have so far received their first dose of vaccine, which includes "more than half" of the over-80s and "half of the people in care homes".
The government hopes that the vaccine programme will inoculate the four groups considered most vulnerable to Covid-19 by mid-February, allowing it to begin lifting some lockdown restrictions.
But he said lockdown will not be lifted in a "great open sesame" and warned that the UK will be dealing with the consequences of the coronavirus pandemic for a "long time"
Asked about his previous promise of achieving normality by spring, Mr Johnson said: "I do think that things will be very different by spring."
He added: "That doesn't mean we're not going to be living with the consequences of the arrival coronavirus pandemic for a while to come - the economic consequences and of course the threat to our health as well.
"We'll have to remain vigilant about this for a long time."
He defended a move to vaccinate groups considered less vulnerable after over-70s and those considered clinically extremely vulnerable were offered jabs for the first time on Monday at sites which have capacity and spare supply.
"Those groups [over 80s and care home residents] remain our top priority, they're an absolute priority for us, but it's right as more vaccine comes on stream to get it into the arms of the other groups on the JCVI list," he said.
He defended the vaccine programme against accusations it's based on a postcode lottery, after Work and Pensions Secretary Therese Coffey complained about jabs in her constituency being offered to some over 70s ahead of over 80s.
Mr Johnson said: "I think actually the whole of the UK is going very well. And, overall, the pace of the rollout is very encouraging.
"We're going as fast as we can but I stress we can do everything we can to open up but when we come to February 15, and the moment when we have to take stock of what we've achieved, that's the time to look at where the virus is, the extent of the infection and the success that we've had.
"It's only really then that we can talk about the way ahead and what steps we can take to relax.
"I'm afraid I've got to warn people it will be gradual, you can't just open up in a great open sesame, in a great bang, because I'm afraid the situation is still pretty precarious."
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