Confirmed cases were higher than 50,000 for the fifth day in a row when UK figures were released on Saturday with a record-high of 57,725 lab-confirmed cases and another 445 deaths within 28 days of a positive test.
Gavin Williamson confirmed on Friday that all London primary schools will remain shut to most pupils next week - rather than just those in certain boroughs as set out earlier in the week - but teaching unions say all schools should close for the next two weeks.
On Saturday evening, the Department for Education said remote learning was "a last resort" and classrooms should reopen "wherever possible" with appropriate safety measures to help mitigate the risk of transmission.
"As we've said, we will move to remote education as a last resort, with involvement of public health officials, in areas where infection and pressures on the NHS are highest," the spokesperson said.
Hundreds of new vaccination sites are due to be up and running this week as the NHS ramps up its immunisation programme with the newly approved Oxford University and AstraZeneca jab.
Some 530,000 doses of the vaccine will be available for rollout across the UK from Monday and more than a million patients have already had their first dose of the Pfizer/BioNTech vaccine which was the first to be approved by the Medicines and Healthcare products Regulatory Agency.
But Ofsted chief inspector Amanda Spielman said children's education cannot be "furloughed" for months while vaccinations are rolled out and time absent from the classroom should be kept to an "absolute minimum", the Sunday Telegraph reported.
Labour leader Sir Keir Starmer said the vaccine roll-out was "our great hope", adding: "I want the Government to throw everything it can at this, harnessing the extraordinary talents of our NHS so we can be vaccinating at least two million Brits a week by the end of the month."
But, writing in the Sunday Mirror, he criticised "a chaotic last minute U-turn on schools", adding: "Confusion reigns among parents, teachers and pupils over who will be back in school tomorrow and who won't."
General secretary of the National Education Union (NEU), Mary Bousted, said schools should stay closed for two weeks to "break the chain" of transmission and prevent the NHS becoming overwhelmed."
The union, which represents the majority of teachers, has advised its members it is not safe to return to classrooms on Monday.
NAHT general secretary Paul Whiteman said the union had started preliminary steps in legal proceedings against the Department for Education, asking it to share its scientific data about safety and transmission rates.
Unions have also called for the reopening of schools in Wales next week to be delayed with Laura Doel, director of school leaders' union NAHT Cymru, saying "the latest data shows that in large parts of Wales, control of infection has been lost".
From January 4, all London primary schools will be required to provide remote learning for two weeks to all children except vulnerable children and those of key workers, who will be allowed to attend.
Mr Williamson said the January 1 decision to expand closures to the nine remaining London boroughs and the City of London was a "last resort".
Under the Government's initial plan, secondary schools and colleges were set to be closed to most pupils for the first two weeks of January, while primary schools within 50 local authorities in the south of England, including 23 London boroughs, were also told to keep their doors shut until January 18.
Linda Bauld, a professor in public health at the University of Edinburgh, told the PA news agency that transmission among primary school pupils was "still very limited" while secondary school pupils, particularly older teenagers, can pass on the virus in the same way as adults.
But health professionals have warned of growing pressure on services with Professor Andrew Goddard, president of the Royal College of Physicians, telling the BBC current case figures are "fairly mild" compared to what is expected in a week's time.
Dr Shondipon Laha, an intensive care consultant in Lancashire, told the Sunday Telegraph: "The situation in hospitals is dire in London, but the situation around the country is only a few weeks behind and London is not at its peak yet.
"If you start seeing London overwhelmed it can happen everywhere else quickly."
The latest figures for Northern Ireland show that 99% of hospital beds are occupied, with only some 30 empty hospital beds left.
Among those receiving hospital treatment after catching Covid-19 is shadow culture secretary Jo Stevens, Labour MP for Cardiff Central.