Boris Johnson is "responding to treatment" for coronavirus as he spends his third day in the hospital.
The prime minister was being kept in St Thomas' Hospital in London "for close monitoring" and remained clinically stable, Downing Street said.
Downing Street said he was not working but could contact those he needed to.
No 10 said a review of lockdown rules would go ahead as planned next week, but the public must "stick with" the measures at what was a "critical time".
A ban on public gatherings of more than two people and the closure of shops selling non-essential goods were among the series of restrictions announced by Mr. Johnson on 23 March to tackle the spread of coronavirus.
Downing Street said a relaxation of the rules would be considered "on or around" the three-week mark on Monday, as was promised by the PM when he introduced the measures.
But health minister Edward Argar said the peak in cases must pass "before we can think about making changes", adding: "It's too early to say when we will reach that peak."
The lockdown in Wales will be extended and not lifted next week, First Minister Mark Drakeford has confirmed.
He said there were signs people staying at home was "having an impact" but "the efforts we are making are not over yet".
The prime minister was admitted to St Thomas' on Sunday, on the advice of his doctor, after continuing to display symptoms of a cough and high temperature 10 days after testing positive for the virus.
Downing Street said Mr Johnson was in "good spirits" as he continued to receive standard oxygen treatment. He was breathing without any assistance, such as mechanical ventilation or non-invasive respiratory support.
The Queen and other senior royals sent messages to Mr. Johnson's family and his pregnant fiancee, Carrie Symonds, saying they were thinking of them and wished the PM a full and speedy recovery.
Foreign Secretary Dominic Raab, who is deputizing for Mr. Johnson, said on Tuesday he was "confident" the PM would recover from this illness, describing him as a "fighter".
In the latest figures across the UK:
The latest UK-wide figures - which use a different timeframe to those of individual nations - said the number of coronavirus hospital deaths rose to 6,159 on Tuesday - a record increase of 786 in a day compared with 439 on Monday.
However, the government's chief scientific adviser told Tuesday's Downing Street briefing the number of coronavirus cases in the UK "could be moving in the right direction".
Sir Patrick Vallance said it was "possible that we're beginning to see... the curve flattening".
Ahead of a spell of sunny weather forecast in some parts of the UK later this week, Mr. Argar urged people to stay at home "however lovely the weather this Easter weekend".
"If we are, as the statistics appear to show, making a little bit of progress, now's the time to hold to it," he told BBC Breakfast.
Regarding a review of lockdown measures, he said: "We need to start seeing the numbers coming down and that's when you're in the negative.
"That's when you have a sense when that's sustained over a period of time, that you can see it coming out of that. We're not there yet and I don't exactly know when we will be."
According to the government's coronavirus legislation, the health secretary must review the need for restrictions at least once every 21 days, with the first review to be carried out by 16 April.
Mayor of London Sadiq Khan told BBC Radio 4's Today program: "I think we're nowhere near lifting the lockdown.
"We think the peak - which is the worst part of the virus - is still probably a week and a half away."
At Tuesday's coronavirus briefing in Downing Street, Mr. Raab was asked about whether his role deputizing for Mr. Johnson gave him full prime ministerial responsibility.
The foreign secretary said he was standing in for the prime minister "whenever necessary" - including leading the daily meetings of the coronavirus "war cabinet".
Mr. Raab said decisions would be made by "collective cabinet responsibility - so that is the same as before".
"But we've got very clear directions, very clear instructions from the prime minister, and we're focused with total unity and total resolve on implementing them so that when he's back, I hope in very short order, we will have made the progress that he would expect and that the country would expect," Mr. Raab added.
Meanwhile, the first patients have been admitted to the NHS Nightingale Hospital in east London - a temporary facility set up at the ExCel conference center.
The admissions come two weeks after the hospital with a planned capacity of 4,000 was formally announced - although an NHS spokesperson stressed limits had not been reached at other sites in London.
The second NHS Nightingale Hospital, at the National Exhibition Centre in Birmingham, is to be opened on Friday, Downing Street said. It will have capacity for up to 2,000 patients if needed.
The prime minister's official spokesman added a third Nightingale Hospital was expected to open in "the next week or so" in Manchester.
