The PM called on people to join a "heroic and collective national effort" and follow social distancing advice.
The number of people who have died in the UK with coronavirus rose to 233 on Saturday, as cases topped 5,000.
It comes as NHS England plans to write to 1.5 million people most at risk.
Those at-risk people will receive letters or text messages strongly advising them not to go out for 12 weeks to protect themselves, the government said.
They include people who have received organ transplants, are living with severe respiratory conditions such as cystic fibrosis or specific cancers, such as blood or bone marrow.
In a message to the country on Saturday evening, Mr Johnson said: "The numbers are very stark, and they are accelerating.
"The Italians have a superb health care system. And yet their doctors and nurses have been completely overwhelmed by the demand.
"The Italian death toll is already in the thousands and climbing. Unless we act together, unless we make the heroic and collective national effort to slow the spread - then it is all too likely that our own NHS will be similarly overwhelmed."
He added he recognised the government was imposing measures "never seen before either in peace or war" - but said they were essential.
As families prepare to celebrate Mother's Day on Sunday, Mr Johnson said the best single present for mothers was to stay away.
It comes after the government this week told all restaurants, cafes and pubs - as well as some other public spaces like gyms and cinemas - to close.
"This time, the best thing is to ring her, video call her, Skype her, but to avoid any unnecessary physical contact or proximity," the PM said.
"And why? Because if your mother is elderly or vulnerable, then I am afraid all the statistics show that she is much more likely to die from coronavirus, or Covid-19.We cannot disguise or sugar-coat the threat."
On Friday, Mr Johnson was asked at his daily press conference whether he would be visiting his own mother, who is 77. He said he would "certainly be sending her my very best wishes and hope to get to see her".
A Downing Street source later said his contact with his mother on Sunday would be over Skype.
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