The number of people in hospital with the Covid variant now known as Delta is rising but not "very significantly," the boss of NHS Providers has said.
Chris Hopson said many of those in hospital in Bolton - the area worst hit by the variant - were younger, with "very few" fully vaccinated patients.
He said this appeared to show vaccines had "broken the chain" between infection and serious illness.
But a scientist pointed to "concerning signs" of the variant's spread.
Another 5,765 infections were recorded in the UK on Saturday, and another 13 deaths were recorded within 28 days of a positive test.
The recent surge in cases is being partly driven by the increased transmissibility of the Delta variant, which was first identified in India.
Mr Hopson said that in Bolton, people in hospital with Covid were "a lot younger" than patients in earlier stages of the pandemic, which meant there was "less demand on critical care".
He added that there were "very, very few" people in hospital who had had both doses of a Covid vaccine, as they had the "build-up of protection after those jabs".
Mr Hopson said in the latest phase of pandemic, the number of people in hospital in Bolton with Covid had not gone past 50, compared with a peak of 170 in November.
He added: "Infection rates have been increasing in a number of different places. We know that the hospitalisations are increasing, the rates of people coming into hospital in those areas are rising. But they are not rising very significantly."
But he warned that the backlog of care for non-Covid illnesses caused by the lockdown means that even a small rise in the number of Covid patients could see hospitals become overstretched.
Asked if the lifting of all restrictions on social contact on 21 June should go ahead as planned, Mr Hopson said the government's decision has become "significantly more finely balanced" due to the NHS still being busy, the Delta variant's increased transmissibility and because many people are still unvaccinated.
He said that he now suspects that, instead of 21 June being an "all or nothing" day when remaining lockdown restrictions are lifted, there may be some things that will be able to change from that day - but it will be "too high risk" to relax other measures.
Public Health England has said that the Delta variant is now the dominant strain of coronavirus in the UK, with the number of cases confirmed by laboratory analysis rising by 79% over the last week to 12,431.
The most affected area of the country remains north-west England, PHE said, with confirmed cases in Bolton rising by 795 over the past week to 2,149. Blackburn with Darwen has seen a near doubling of cases to 724 in total.
At next week's G7 summit of world leaders, Prime Minister Boris Johnson will call on his counterparts to pledge to vaccinate the world's population against coronavirus by 2022, saying it would be "the single greatest feat in medical history".
It comes a week after MPs and peers wrote to Mr Johnson urging him to commit to donating at least 100 million doses of coronavirus vaccines before the end of the year.
So far, the UK has only vowed to donate surplus shots to the international Covax scheme and on Friday, Health Secretary Matt Hancock said: "At the moment we don't have any excess doses, we're just getting them into arms as quickly as possible."
What might life look like after 21 June?
At the moment, ministers say there is nothing in the current data to suggest that England can't move into step four of the roadmap on 21 June.
But what does the roadmap actually commit to?
It says the government will reopen venues like nightclubs, it hopes to lift caps on numbers at weddings and remove "all legal limits on social contact".
If you look at the rest of the roadmap, it's pretty clear that "social contact" has been used to refer to how many people can gather and where.
So what it doesn't commit to is ending measures such as the "one metre plus" social distancing rule in pubs and restaurants, wearing face masks or even working from home.
Those measures are being looked at by a government review.
The results had been expected to be announced at the end of May but are now likely to come with the decision on step four on 14 June.
That raises the possibility of step four going ahead as planned but with at least some social distancing measures staying in place. We shouldn't expect a return to normal overnight.
Chancellor Rishi Sunak said that "no final decisions had been made" about the June unlocking, as ministers were "looking at the data as it comes in".
He said: "As the prime minister has said we need to make sure that we've vaccinated and protected enough people to give us the reassurance we need given the new variant that is currently in the UK."
Mr Sunak added that it was "too early to say where that will conclude" but that the data would be examined and a decision would be made before 21 June.
Meanwhile, Dr Adam Kucharski, a member of the Scientific Pandemic Influenza Group on Modelling (SPI-M) that advises the government, said there were a "number of concerning signs" following the earlier finding that vaccine first doses aren't as effective against the Delta variant and that people infected with the variant could be more likely to need hospital care.
He told BBC Radio 4's Today programme that the increase in transmission could potentially cause "considerable problems" and that it was "hard to be confident" that it would not lead to a big number of hospital admissions.
Asked for his view about the June lifting of restrictions, Dr Kucharski said the new variant had "changed" the equation.
"We are not facing the virus that we were facing two months ago. If we were facing the B.117 (Kent) variant as the dominant one, the fact it (the vaccines) is working very well, case numbers are coming down nicely, we could have some more confidence that there could be reopening without seeing those surges.
"But we're not... if say in two weeks we're in a situation where hospitalisations have been rising, where local health systems are coming under pressure, I think it will be quite difficult to justify adding more transmission to that kind of situation."
Concerns over the Delta variant mean vaccination is being accelerated in many areas, with second doses being offered earlier to some groups.
In the UK, more than 40 million people (76.2% of adults) have now had their first dose of a vaccine, according to the latest figures. Some 27 million people have had their second dose, equivalent to 51.6% of adults.
This weekend, people over the age of 18 living in Harrow, north-west London, have been invited to have a Pfizer-BioNTech vaccine dose, with long queues seen outside the Belmont Health Centre in Stanmore on Saturday.
A spokeswoman at North West London Integrated Care System (ICS) said: "We continue to prioritise those in eligible groups.
"We made a local clinical decision to invite wider groups today so that available doses could be used without waste and only people who have been invited from our local communities will be vaccinated."
It follows a one-day event at Twickenham rugby stadium on Monday as thousands of people from the local area queued up for first dose jabs.
In England, only people aged 30 and over are currently eligible to book a vaccination online or through the national booking service on 119.
Meanwhile, surge coronavirus testing is being rolled out in parts of Berkshire and Bradford after cases of the Delta variant were identified there.
Everyone over the age of 12 who lives and works in specific postcode areas of Reading and Wokingham are being offered PCR tests for a fortnight from Monday.
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