"Now is not the time to be complacent," NHS England medical director Stephen Powis has warned at the government's latest coronavirus press conference.
Prof Powis said "every one of us" had a part to play if the UK was to keep the death toll under 20,000.
Business Secretary Alok Sharma also announced changes to insolvency rules and the NHS supply chain.
The number of people to have died with coronavirus in the UK has now reached 1,019.
Mr Sharma said insolvency rules would be changed to allow firms greater flexibility as they faced the coronavirus crisis.
He added that a range of measures to boost the supply of personal protective equipment, such as face masks to protect frontline NHS staff, were also being introduced.
"Red tape" would be reduced to allow new producers of hand sanitiser to bring products to market "in a matter of days", he said.
Prof Powis was also asked about the spread of the disease after a further 260 UK deaths in 24 hours.
He replied: "Clearly there has been a big increase in deaths today.
"If we can keep deaths below 20,000 we will have done very well in this epidemic," Prof Powis added, echoing comments from the government's chief scientific officer earlier in the month.
He continued: "It is early at the moment and the scientists who are working with the government to model what we can expect are of course adjusting their predictions now as we start to see the actuality of the epidemic in the UK, rather than what we believed might have happened a few weeks ago."
By BBC health reporter Philippa Roxby
Despite today's news of the largest rise in people dying in the UK with Covid-19, the message is that everything we are all doing now to reduce social contact is having an impact - we just can't see it in the figures yet.
That's because it takes time for the effects of not going out and not going to work to be felt in hospitals, where the most seriously ill are being tested and some are, sadly, dying.
The epidemic is expected to peak in the UK in the next two to three weeks.
Until then, it is likely the number of people confirmed to have the coronavirus and the number of people dying after testing positive for the virus will continue to grow at a similar pace - doubling every two or three days.
The hope is that everyone's efforts now to stay at home and reduce the spread of this deadly virus will slow that growth rate at the right time, and, in time, bring it down.
Every one of us now has a part to play in making that happen.
Explaining the new measures to help businesses, Mr Sharma said he hoped the changes would help firms "emerge intact the other side of the Covid-19 pandemic".
"These measures will give those firms extra time and space to weather the storm and be ready when the crisis ends, while ensuring creditors get the best return possible in the circumstances," he said.
The new rules will allow companies undergoing restructuring to continue to get access to supplies and raw materials.
And, he said, there would be a temporary suspension of "wrongful trading" rules for company directors to remove the threat of personal liability during the pandemic, which will apply retrospectively from 1 March.
We've seen the government announcing measures to give businesses access to cash in order to keep going over the past few weeks.
Now it has turned to those that are facing insolvency.
It has announced a temporary suspension of "wrongful trading laws", which stop company directors from being prosecuted for keeping a business going when it's insolvent or hasn't got a reasonable prospect of survival.
The idea is to remove the fear bosses might have of trading through such turbulent and uncertain times and to give them breathing space. The chance to "bounce back", as the business secretary put it.
However, much of the detail still needs fleshing out - such as how the government intends to keep suppliers working with companies that are in financial trouble.
While business groups have welcomed the move, the trade association for the UK's insolvency industry, R3, believes this change could be open to abuse.
All agree that this will stop a sudden spike in the number of companies going under - but in the long run it won't stop the collapse of businesses that fundamentally aren't viable.
Asked how the prime minister - who is self-isolating with mild symptoms having tested positive for the virus - was getting on, Mr Sharma said he continued to lead the government's effort in combating Covid-19
"This morning he held a video conference call and he will continue to lead right from the front on this."
He added that this showed why they were asking people to follow the government advice in terms of staying at home.
Earlier, it was announced frontline hospital staff in England were starting to be tested to see whether they have coronavirus.
Workers with symptoms and those who live with people who have symptoms will be checked - starting with hundreds of critical care doctors and nurses.
Tests for A&E staff, paramedics and GPs are expected to follow, then social care staff.
In Wales and Scotland, health workers are already being tested. Northern Ireland's testing begins on Monday.
The British Medical Association said the move towards testing NHS staff in England was "long overdue", following concerns that healthy members of staff may have been self-isolating at home when they did not need to.
The new tests being offered to frontline NHS staff are antigen tests, which indicate if someone is currently infected and risks spreading it to others.
Another type of test, called the antibody test, could indicate whether someone has recently had the virus. It is not available to the public yet but Public Health England is ordering it in, for use once it is proven to work.
In other developments:
- Secretary of State for Scotland, Alister Jack, is self-isolating after he developed coronavirus symptoms. He is said to be experiencing mild symptoms but has not been tested for Covid-19
- Prime Minister Boris Johnson chaired the government's morning Covid-19 by video-link as he isolates in Number 11 after testing positive for coronavirus and self-isolating.
- Photos showed work under way to turn London's Excel centre into a temporary hospital with capacity for 4,000 people
- Two new temporary hospitals will be set up in Birmingham and Manchester to help the NHS cope with the virus
- The Local Government Association says council workers are being physically and verbally abused for implementing the government's social distancing policy. Workers have been spat at, sworn at and racially abused, it said
- The Mayor of Greater Manchester, Andy Burnham, said he had received about 300 complaints from employees and the public about businesses not following social distancing guidelines.