Covid passes will also be needed to get into nightclubs and large venues from next week.
Boris Johnson announced the government was moving to its back-up plan of extra Covid rules at a news conference.
"It's not a lockdown, it's Plan B," the prime minister said.
He said moving to the tougher measures was the "proportionate and responsible" thing to do.
Mr Johnson said more is still being learned about new variant Omicron and the picture might get better, but that it "could lead to a big rise in hospitalisations and therefore sadly in deaths".
He said the new variant was "growing much faster" than Delta and early analysis suggested cases could be doubling every 2.5 to three days.
Under the new rules:
From Friday, face masks will be required in more public settings - including theatres and cinemas
From Monday, people will be asked to work from home where possible
From Wednesday, the NHS Covid Pass will also be required for visitors to nightclubs, indoor unseated venues with more than 500 people, unseated outdoor venues with more than 4,000 people and any event with more than 10,000 people
The NHS Covid Pass crashed hours after the PM's announcement, with users unable to download their domestic or travel passes. NHS Digital said it was investigating the issue as a priority.
Mr Johnson said Christmas parties and nativity plays should still go ahead - as long as the guidance is followed.
"The best way to ensure we all have a Christmas as close to normal as possible is to get on with Plan B," he said. "Irritating though it may be, it is not a lockdown."
Ministers have repeatedly said there are no plans for another lockdown in England.
How big a threat is Omicron?
What are the Plan B rules?
What's the new working from home guidance?
Where will I need a Covid passport and how do I get one?
Many of the questions the PM faced centred on the row over the Downing Street Christmas party at the height of lockdown rules last December.
Government adviser Allegra Stratton - who was seen with other No 10 staff joking about the party in a leaked video from last year - resigned just before the news conference, saying she would always regret her remarks.
Earlier Mr Johnson apologised in the Commons for this video, although he said that he had been repeatedly told there had been no party. The Metropolitan Police has now said they will not investigate the issue, due to lack of evidence.
The prime minister denied that the Plan B announcement was timed to divert attention from the Christmas party, saying the government did not want to delay bringing in the rules designed to protect public health.
Laura Kuenssberg challenges PM over new restrictions
There are currently 568 confirmed cases of Omicron in the UK, figures show - although the UK Health Security Agency estimates that the true number of infections is about 20 times higher and probably closer to 10,000.
Other nations of the UK - which are in charge of their own Covid rules - have already brought in stricter restrictions similar to Plan B.
People in Wales and Scotland have already been told to work from home where possible and Northern Ireland recently strengthened its advice.
Covid passes are also currently required for venues in Scotland, Wales and Northern Ireland.
In England, people will be exempt from showing their Covid passport when attending religious worship, weddings and funerals.
Working from home: 'It will be painful to go into reverse gear again'
Covid passports 'devastating' for nightclubs
How worried should we be about Omicron?
What are England's mask rules now?
Scotland's First Minister Nicola Sturgeon urged people to follow the rules, adding: "Even if you feel angry with a politician just now, please remember just how important compliance is for the health and safety of you, your loved ones and the country."
Sir Patrick Vallance - the government's chief scientific adviser - also urged people to comply. "It only works if we all do it," he said.
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Analysis box by Nick Triggle, health correspondent
The measures introduced are not about stopping the spread of Omicron, or even reducing the number of cases.
Instead, it is about slowing the spread. The big concern is a huge surge of cases that leads to a wave of hospital admissions that could overwhelm the NHS.
There are suggestions Omicron is leading to milder illness. There is logic in that as reinfections, or infections after vaccination, are less likely to lead to serious illness.
But if infections rise quickly enough there is still the risk of hospital cases rising. A variant that causes half the amount of serious illness will lead to more people ending up in hospital if infection rates more than double.
So the hope is that these measures will flatten and delay the peak. That will allow more boosters to go into arms - and the evidence that is emerging on vaccine effectiveness against the new variant suggests that will make an important difference to the spread of the virus.
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There will be a debate and vote in the House of Commons next week on the new rules, said Health Secretary Sajid Javid, who made a statement to MPs at the same time as Mr Johnson's news conference.
After Mr Javid's statement, several Conservative MPs expressed their dismay at the introduction of tougher restrictions. Tory MP Dr Liam Fox said: "We cannot allow permanent threats of overloading the NHS as a means to maintain semi-permanent restrictions on our people."
The health secretary also said vaccine manufacturers may have new vaccines ready to trial "within weeks" to combat Omicron. However, existing vaccines should still protect people against severe illness from Omicron, the World Health Organization said.
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