As the number of first doses delivered in the UK passed four million, ministers and officials said it was still too soon to rely on the jab “coming to our rescue”.
The warning came as the latest official figures showed there was a record 37,475 people in hospital with the disease across the UK.
As the pressure continues to mount, the NHS Confederation has said the health service could reach its limit for critical care beds this week.
Meanwhile scientists advising the Government have warned there is a danger that people could start relaxing their guard as the vaccine started to become available.
The latest minutes of the Scientific Advisory Group for Emergencies (Sage), released last week, called for close monitoring of the situation with a system of “rapid alerts” if adherence to the rules begins to fall off.
“There is a risk that changes in behaviour could offset the benefits of vaccination, particularly in the early months of vaccine rollout,” it said.
At a No 10 news conference on Monday, Health Secretary Matt Hancock issued a direct appeal to the public, saying: “Don’t blow it now.”
NHS England’s medical director Professor Stephen Powis said that with around 15,000 hospital admissions with coronavirus since Christmas Eve, the pressures on the service would continue for some time to come.
“It is absolutely critical that we continue to stick to those social distancing rules that are in place. That we don’t rely yet on vaccines coming to our rescue,” he said.
The Government says it is on track to vaccinate around 15 million high-priority people across the UK by February 15, including frontline health and social care staff, the over 70s and people in care homes.
Once those vaccines have taken effect, around two to three weeks later ministers will consider whether lockdown measures can be eased in England.
Despite pressure from Tory MPs to move as quickly as possible, Boris Johnson has warned there will be no “open sesame” moment when restrictions will all be lifted together.
Speaking during visit to a manufacturing facility for the Oxford/AstraZeneca vaccine on Monday, the Prime Minister said the UK was still in a “pretty precarious” position and that any loosening would be gradual.
Mr Hancock acknowledged that some areas of the country had made better progress than others but said they were putting more supplies of the vaccine into those that were falling behind.
“What we’re doing now is making sure that whilst they, of course, will be able to move onto the next group, we’re prioritising the supply of the vaccine into those parts of the country that need to complete the over-80s,” he said.
“But we don’t want to stop the areas that have effectively done that job already, we want them to carry on, but the priority of the vaccine is according to the JCVI (Joint Committee on Vaccination and Immunisation) prioritisation list.
“The critical thing is to make sure that everybody can get it, that we’re putting more supply into the areas that have got more to do.”
Wales has faced criticism in the past week for vaccinating fewer people in proportion to its population than the other home nations.
As of Saturday, 4% of the population in Wales had been vaccinated, compared with 4.1% in Scotland, 5.9% in England and 7.4% in Northern Ireland.
First Minister Mark Drakeford dismissed the statistics as “very marginal differences” and insisted Wales was “on track” to vaccinate the top four priority groups by the middle of February, with almost 152,000 having received their first injection.