The UK is "at a dangerous moment" and the easing of lockdown "has to go slowly", England's deputy chief medical officer has said.
Prof Jonathan Van-Tam said the consensus among scientists was that the new measures were not expected to push the rate of infection above 1.0.
However, he cautioned the public to be "sensible and proportionate with the freedom we have wanted to give people".
"Don't tear the pants out of it," he told the daily Downing Street briefing.
It comes as scientific advisers to the government warned of the risk of lifting the lockdown in England.
From Monday, schools will reopen and up to six people can meet in England, with other nations also easing measures.
In Scotland, people can now meet others from one other household - up to a maximum of eight people - as long as it is outdoors; in Northern Ireland, the maximum number of people who can meet outdoors is six.
In Wales, people will be allowed to meet any number of people from two households outdoors from Monday, but beauty spots will remain closed.
Professor John Edmunds, a member of Sage - the Scientific Advisory Group for Emergencies which advises the government - said the levels of coronavirus were still "very high", adding that it was a "political decision" to ease measures.
Estimates by the Office for National Statistics suggest there are currently 8,000 cases per day in England alone.
Fellow Sage member Sir Jeremy Farrar said the NHS test and trace system should be "fully working" before measures were introduced.
Two further members of Sage, Professors Peter Horby and Calum Semple, also spoke out about the relaxation of lockdown measures, suggesting it was still "too early".
Culture Secretary Oliver Dowden, who led the briefing on Saturday, said scientists were "absolutely right to urge caution".
He said it was a risky moment, but described the measures agreed so far as baby steps.
"At each stage, we are not going to proceed unless we are confident that we can do so in a safe way," he said.
He added that there were more than 50 scientists in Sage "all of whom will have their different perspectives" and the government takes a "collective view".
A further 215 people across the UK who tested positive for coronavirus have now died, taking the total death toll to 38,376.
This is a significant moment.
Throughout the pandemic, the scientists and politicians have been following the same script.
Now there is a clear, loud and public split between some of those advising government and those "following the science".
They may not be household names, but those speaking out are big figures in their fields and sit on Sage for a reason.
On Thursday, Boris Johnson confirmed the relaxing of lockdown. At the same televised briefing, the PM's chief science advisor, Sir Patrick Vallance, warned there was "not a lot of room" for manoeuvre and the data "urges caution".
This is largely a disagreement about "when" to start lifting lockdown rather than "how".
If you go early then we all get a bit of our lives back sooner, but it means we will stay close to that 8,000 cases a day figure and have less time to prevent the NHS being overwhelmed if a second wave comes.
If you wait longer, then cases are much lower and the virus easier to control, but the price is keeping lockdown and all the pain it is causing.
Prof Van-Tam told the press conference that the lockdown easing must go "painstakingly" slowly, adding: "The scientists will continue to give that advice to the government. No apologies for that - we will absolutely continue to do that."
"I believe this is also a very dangerous moment. We have to get this right."
He said Sage was clear that the chances of the new measures allowing the infection rate - R - to go above 1.0 were slim "with good compliance and the test and tracing system in place".
"Our science advice has been - so far - that the social distancing that has already been eased is consistent with keeping the R below one."
But he stressed the "dual responsibility" of the government and the public, whom he urged to "actually follow the guidance".
Asked about the public's ongoing adherence to those measures and the row over government adviser Dominic Cummings' decision to travel to County Durham during lockdown, Prof Van-Tam said: "In my opinion the rules are clear and they have always been clear. In my opinion they are for the benefit of all and they apply to all."
Prof Van Tam agreed to respond to a question on the issue, something other advisers had previously refused to do.
The culture secretary said Mr Cummings had given a "very extensive explanation of his behaviour".
It comes as Mr Dowden announced that competitive sport will be allowed behind closed doors in England from 1 June, paving the way for the first live action in almost three months.
"The British sporting recovery has begun," Mr Dowden told the briefing. "Football, tennis, horse racing, Formula One, cricket, golf, rugby, snooker and others are all set to return to our screens shortly."
It was also announced that people in England will be able to exercise outside with up to five others from different households from Monday. People must remain at least 2m apart and physical contact sessions are still banned.
In other developments:
- The Premier League says it is "thrilled" to be given the go-ahead by the UK government to resume behind closed doors
- The level of testing is staying "very" steady, at around 120,000 tests a day, says Prof Van-Tam
- Some vulnerable people have been told they have been removed from shielding lists via text message, without the knowledge of their GP
- Tory MPs call for change in 2m social distancing rule
- Renters are more likely to be struggling with payments than those who own their own homes, research from the Resolution Foundation suggests
- Restaurant employees at Debenhams say they will be made redundant at the end of the month as the chain's struggles continue