The PM acted “decisively” in announcing a new lockdown in England “in the face of new information”, Rishi Sunak says.
People must now stay at home except for a handful of permitted reasons and schools have closed to most pupils.
The chancellor said the action was “regrettable” but it was “right we take these measures”, which will be reviewed on 15 February, to suppress the virus.
It came after UK chief medical officers recommended the Covid threat level be increased to five – its highest level.
The prime minister said vaccinating the top four priority groups by mid-February could allow restrictions to be eased.
Tough new lockdown restrictions forbidding people from leaving home for non-essential reasons have also come into force across the Scottish mainland – these rules will be reviewed later this month.
In Wales, which has been in a national lockdown since 20 December, schools and colleges will stay shut until 18 January for most pupils.
Northern Ireland, which entered a six-week lockdown on 26 December plans to put its stay-at-home message into law, and will have an “extended period of remote learning”, the Stormont executive said.
The UK reported a record 58,784 cases on Monday, as well as a further 407 deaths within 28 days of a positive test.
Meanwhile, Prime Minister Boris Johnson is due to hold a press conference in Downing Street at 17:00 GMT with chief medical officer for England Prof Chris Whitty and the government’s chief scientific adviser, Sir Patrick Vallance.
Elsewhere, Mr Sunak has announced that businesses in the retail, hospitality and leisure sectors are to be given a one-off grant worth up to £9,000, with the measure costing £4bn across the UK.
Cabinet Office minister Michael Gove told BBC Breakfast: “The four chief medical officers of the United Kingdom met and discussed the situation yesterday and their recommendation was that the country had to move to level five, the highest level available of alert that meant there was an imminent danger to the NHS of being overwhelmed unless action was taken.
“And so in the circumstances we felt that the only thing we could do was to close those primary schools that were open.”
He said the action was taken “with the heaviest of hearts” and “we had to act” following that advice. ” With a heavy heart but with clear evidence we had to act.”
Mr Gove also confirmed GCSE and A-Level exams in England were being cancelled this year, saying Education Secretary Gavin Williamson would make a statement in the House of Commons on Wednesday on “how we want to make sure children are fairly assessed”.
“It is a very, very difficult time for the whole country, that’s why it’s so important we do everything we can in government to vaccinate people,” he said.
He said a million people had been vaccinated so far “up until the weekend” and it was hoped that number would reach more than 13 million in February.
When asked about the target of two million vaccines a week and concerns over logistics and the safety systems, he said: “The process of making sure that the vaccines can be placed in the appropriate vials and then safely injected into people’s arms is a complicated exercise, but the NHS has more than risen to the challenge.”
The government was “looking at further options” to restrict international travel, he said.
Mr Gove told Sky News he could not say exactly when the lockdown in England would end, adding: “I think it is right to say that as we enter March we should be able to lift some of these restrictions but not necessarily all.”
Prof Andrew Hayward – a member of the government’s Scientific Advisory Group for Emergencies (Sage) – told BBC Radio 4’s Today programme the lockdown measures “will save tens of thousands of lives”.
But he said “the virus is different” and “it may be that the lockdown measures that we have are not enough”
“This lockdown period we need to do more than just stay at home, wait for the vaccine, we need to be actively bearing down on it,” he said.
But it doesn’t mean that a national instruction to close the doors was automatic. Or indeed that new lockdowns in England and Scotland aren’t still dramatic and painful.
With tightening up in Wales and Northern Ireland too, the spread of coronavirus this winter has been faster than governments’ attempts to keep up with it – leaving leaders with little choice but to take more of our choices away.
There is much that’s an echo of March. Work, school, life outside the home will be constrained in so many ways, with terrible and expensive side-effects for the economy.
This time, it’s already spluttering – restrictions being turned on and off for months have starved so much trade of vital business.
But there’s a lot that’s different too. After so long, the public is less forgiving of the actions taken, and there is frustration particularly over last-minute changes for schools; fatigue too with having to live under such limits.
Announcing England’s lockdown on Monday, Mr Johnson said hospitals were under “more pressure from Covid than at any time since the start of the pandemic”.
He ordered people to stay indoors other than for limited exceptions – such as essential medical needs, food shopping, exercise and work that cannot be done at home – and said schools and colleges should move to remote teaching for the majority of students until at least half term.
And the prime minister said all care home residents and their carers, everyone aged 70 and over, all frontline health and social care workers, and the clinically extremely vulnerable will be offered one dose of a vaccine by mid-February.
While the rules become law in the early hours of Wednesday, people should follow them now, Mr Johnson added.
Under England’s new measures, support and childcare bubbles will continue, and people can meet one person from another household for outdoor exercise.
Exercise should be limited to once per day.
Communal worship and funerals can continue, subject to limits on attendance. Weddings are allowed in “exceptional circumstances” with up to six people.
Mr Johnson said the new variant of coronavirus, which is up to 70% more transmissible, was spreading in a “frustrating and alarming” manner and warned that the number of Covid-19 patients in English hospitals is 40% higher than the first peak.
People who are clinically extremely vulnerable will be contacted by letter and should now shield once more, Mr Johnson said.
The House of Commons has been recalled to allow MPs to vote on England’s new restrictions on Wednesday.
Labour leader Sir Keir Starmer said his MPs would “support the package of measures”, saying “we’ve all got to pull together now to make this work”.
Earlier on Monday, First Minister Nicola Sturgeon issued a stay-at-home order for Scotland, beginning at midnight and lasting until the end of January.