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.
Boris Johnson said vaccinating the top four priority groups by mid-February could allow restrictions to be eased, with Cabinet Office minister Michael Gove telling Sky News the measures may remain until March.
Meanwhile, the prime minister 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.
Tough new lockdown restrictions forbidding people from leaving home for non-essential reasons have also come into force across the Scottish mainland. Wales has been in a national lockdown since 20 December and Northern Ireland entered a six-week lockdown on 26 December.
The UK reported a record 58,784 cases on Monday, as well as a further 407 deaths within 28 days of a positive test.
Mr 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.
"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, Mr Gove said the vaccination process was 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."
Cabinet Office minister Michael Gove saying the lockdown may have to last to March may not come as much of a surprise to many.
While the government has set a target of offering the most at-risk a jab by mid February, it will take several weeks longer for the full effect to be felt given it takes time for an immune response to kick in.
The bigger question is whether or not the government could have acted earlier.
It was clear before Christmas the new variant was pushing up infection rates - and that in turn would mean more hospital admissions.
The delay looks costly. Since Christmas Day, the number of Covid-19 patients in hospital has risen by 50% alone - enough to fill 18 hospitals.
While the government did introduce tier four the weekend before Christmas in parts of the south east of England, which banned mixing over the festive period and led to the closure of non-essential shops and gyms, most of the country were allowed to meet up on Christmas Day.
Infections from Christmas Day are now being felt - the numbers have been rising sharply ever since. Some of these are next week's hospital admissions - and is why the chief medical officers warned of the risk of hospitals becoming overwhelmed, which Mr Gove said persuaded them to act on Monday.
If lockdown had come earlier, it may well have been shorter.
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.
At Scotland's daily briefing, First Minister Nicola Sturgeon called for people to hold on to the fact there was now "a clear route out of this pandemic".
She said there had been urgent discussions between the four home nations about whether border controls should be tightened - and she hoped there would be an announcement soon.
In other developments:
- Chancellor Rishi 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
- Mr Gove has confirmed GCSE and A-Level exams in England are being cancelled this year and Education Secretary Gavin Williamson is due to make a statement in the House of Commons on Wednesday
- First Minister Nicola Sturgeon has said the Scottish NHS hopes to vaccinate everyone over the age of 50 and younger people with underlying health conditions by the start of May
- In Northern Ireland, which entered a six-week lockdown on 26 December, post-primary transfer tests have been cancelled.
- Train services are expected to be reduced during the national lockdown in England
- MPs have been told that some of the UK's biggest music festivals could be called off by the end of this month
- Travel firms Tui and Thomas Cook have cancelled holidays after new coronavirus lockdown restrictions were imposed
- Buckingham Palace has announced that there will be no royal garden parties this year - in either London or Edinburgh
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.
People who are clinically extremely vulnerable will be contacted by letter and should now shield once more, Mr Johnson said.
While the rules become law in the early hours of Wednesday, people should follow them now, Mr Johnson added.
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.
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".
At-a-glance: New rules
- People cannot leave their homes except for certain reasons, like the first lockdown last March
- These include essential medical needs, food shopping, exercise and work for those who cannot do so from home
- All schools and colleges will close to most pupils from Tuesday with remote learning until February half term
- Early years settings such as nurseries will stay open
- End-of-year exams will not take place this summer as normal
- Elsewhere, university students should not return to campuses and will be taught online
- Restaurants can continue to offer food delivery, but takeaway alcohol will be banned
- Outdoor sports venues - such as golf courses, tennis courts and outside gyms - must close
- But outdoor playgrounds will remain open
- Amateur team sports are not allowed, but elite sport such as Premier League football can continue
In mainland Scotland and Skye:
- Nursery, primary and secondary schools will close to all most pupils until February. Learning will move online
- People should only leave home for essential reasons, like the first lockdown last March
- Those who are shielding should not go into work, even if they cannot work from home
- A maximum of two people from up to two households can meet outdoors, excluding under-12s who can play together outside
- Places of worship will close except for weddings (up to five people) and funeral services (up to 20 people). Wakes are not allowed
- The definition of an essential business will be tightened with premises such as ski centres, large retail showrooms, and cosmetic clinics required to close
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