The UK on Sunday reported 54,990 confirmed cases of the coronavirus and 454 more deaths, government figures show.
And British Prime Minister Boris Johnson said on Sunday that harsher Covid-19 restrictions could be on the way.
Concerns that the tier system may not be enough to bring the virus back under control meant restrictions might be about to get tougher, Mr Johnson told the BBC.
“There are obviously a range of tougher measures that we would have to consider,” he said.
“I’m not going to speculate now about what they would be.”
Despite this, Mr Johnson insisted schools were safe and advised parents to send their children in to classes in areas where rules allowed it.
“There is no doubt in my mind that schools are safe and that education is a priority,” he said.
Much of England is already living under the toughest level of restrictions under a four-tier system designed to stop the spread of the virus and protect the national healthcare system.
The UK government’s response has been heavily criticised.
But vaccination is set to accelerate on Monday, with the first 530,000 doses of the newly approved drug developed by the University of Oxford and AstraZeneca ready to be administered, Mr Johnson said.
He said he hoped tens of millions would be treated within the next three months.
The UK’s largest teachers’ union said schools in England must stay closed for at least two weeks after the Christmas holiday.
Some local authorities and unions oppose reopening schools and have threatened to act against government advice.
Others say closures have a harmful effect on pupils.
"We must renew and maintain the consensus that children's time out of school should be kept to the absolute minimum," Amanda Spielman, the chief schools inspector, wrote in The Sunday Telegraph.
The National Education Union said it was unsafe for teachers to be in schools because of inadequate social distancing and ventilation, and a lack of personal protective equipment.
The reopening of most secondary schools in England has already been delayed by two weeks.
Most primary school pupils in London will be taught remotely next week, a decision described as a “last resort” by Education Minister Gavin Williamson, but the government is being called on to expand the measures.
Mary Bousted, the joint secretary of the National Education Union, said all schools should be closed to stop the country’s hospitals being overwhelmed.
“We know that pupils now can transmit the virus through their homes, through to their families and into the community,” Ms Bousted told the BBC.
“They’re the most effective transmitter of the virus.
“You combine that with the new variant being up to 70 per cent more infective than the previous Covid virus, which was very infective, then it’s clear we have to do something to break the chain of rising levels of infection in our community.”