Parks in part of London are being shut after criticism of large numbers of tourists visiting beaches and beauty spots.
Mayor of London Sadiq Khan urged people to "stop social mixing", saying "people will die" if they don't.
Authorities in the Yorkshire Dales and Lake District asked people to stay away, saying "now is not the time for tourism".
Mr Khan said people should not leave home "unless you really have to".
Hammersmith and Fulham council will close parks from Sunday night while the Royal Parks, responsible for Hyde, Regent's and St James' Parks, are closing kiosks and cafes.
Roads to outer parks - including Richmond, Bushy and Greenwich Parks - will be closed, with the Royals Parks calling social distancing "absolutely crucial".
"If people do not follow social distancing guidelines, we will have no choice but to consider closing the parks," the body said.
Latest figures show just less than 2,000 people in London have been infected with coronavirus with 93 deaths, and Mr Khan told BBC One's the Andrew Marr Show the capital was "weeks ahead of the rest of the country".
Asked if the Tube should be closed completely, Mr Khan said he was keen to keep some trains running so "critical workers" could get to work.
"Nobody else should be using public transport," he added.
We’ve closed our parks and gardens to restrict the spread of coronavirus, as well as, built properties. Many of our car parks for countryside and coastal locations are now likely to be closed. We urgently request people to stay local, observe social distancing and to not travel. pic.twitter.com/FIuP9l744a— National Trust (@nationaltrust) March 22, 2020
The National Trust shut parks and gardens over the weekend and said countryside and coastal car parks were "now likely to be closed".
"The number of people coming to the area and acting so irresponsibly at a time of national crisis cannot be acceptable," he said.
"If people chose to come here and ignore government advice regarding social distancing, then I would suggest they do not travel to the Yorkshire Dales at all and stay at home."
Residents of the Dales have also condemned some of the visitors and suggested the authority close its car parks.
One said: "'Avoid unnecessary travel' means precisely that. A trip to the coast, the Yorkshire Dales or wherever is certainly not necessary travel.
"Lockdown will eventually happen if people continue to think 'it doesn't apply to me'."
Anthony Bishopp, the mayor of Hunstanton, Norfolk, said Saturday was "ridiculous" with people queuing close together to get fish and chips.
Some residents went out at the end of the day to clean cash machines and railings after the influx.
Jessica Stevenson said it was "like a bank holiday" in the Derbyshire Peak District village of Matlock Bath.
Cumbria Police and Cumbria County Council are asking visitors to stay away from the Lake District to limit the spread of the coronavirus, saying: "Now is not the time for tourism."
"Now that pubs, restaurants, cafes and non-essential shops and visitor attractions have been advised to close, the Lake District is no longer conducting business as usual," a police spokesman said.
Assistant Chief Constable Andrew Slattery said: "I must urge people living outside the county not to visit.
"A national emergency shut-down of businesses and schools is not an excuse for a holiday.
"The health, social care and emergency services in Cumbria are resourced to serve the 500,000-resident population and will be stretched to breaking point by this crisis.
"Large numbers of visitors will only place an additional burden on these hard-pushed professionals."
The Whitstable Oyster Company has apologised for opening its beachside takeaway premises on Saturday saying all takings would be donated to the National Emergencies Trust coronavirus appeal.
"It was certainly not our intention to play a part in encouraging or facilitating the gathering together of people," the company said.
But some councils have said it is not yet clear how social distancing can be enforced.
"The council's powers are limited in these circumstances so we are working urgently with the police on what action can be taken," a spokesman for Canterbury City Council said in response to "deplorable" visitor numbers at Whitstable beach.
"We all need to work together to fight this virus and common sense is one of our biggest weapons," the spokesman said, adding: "People should follow the government's advice both to the letter and in the spirit in which it is intended."
Residents in Devon and Cornwall have also been asking people to stay away.
One said: "Sorry and all that [but] please do not come here where we do not have the capacity to mop up anything you may bring with you".
Another said: "You can come visit when things are back to normal."
Reports of people arriving at holiday lets and second homes in places such as Salcombe in Devon - plus people parking at popular spots in both counties, including Dartmoor and Cornwall's beaches - have been causing tension on social media.
The mayor of Salcombe, Niki Turton, told the BBC: 'We can't really stop them coming.
"But we would wish that they would do as we are doing - staying at home staying safe, protecting the vulnerable and just not putting extra strain where it's not needed."
Visit England has suggested people enjoy attractions from their own homes by visiting online museum archives or watching movies and TV shows filmed at beauty spots.
We know it is a difficult time and we can’t do some of the usual activities we love, such as trips away with friends and adventures across the country. So we thought we’d switch things around and bring England into your homes instead! pic.twitter.com/YzZUUrighU— VisitEngland (@VisitEngland) March 18, 2020