But Mr Johnson said the Indian variant of Covid-19 could "pose a serious disruption" to move to Step 4 in June as he announced planned easements of Step 3 of the Government's roadmap will go ahead on Monday, 17 May.
He also urged people "to exercise the greatest caution because the choices we each make in the coming days will have a material effect" on England's progress.
The Prime Minister said that "at this stage there is no evidence of increased cases translating into unmanageable pressures" on the NHS.
But Mr Johnson has said that remaining second doses for the over-50s will be accelerated so they come eight weeks after the first.
Speaking at a Downing Street press conference, the Prime Minister said: "I believe we should trust in our vaccines to protect the public whilst monitoring the situation as it develops very closely because the race between our vaccination programme and the virus may be about to become a great deal tighter and it's more important than ever therefore that people get the protection of a second dose.
"So following advice from the Joint Committee on Vaccination and Immunisation we will accelerate remaining second doses to the over-50s and those clinically vulnerable right across the country so those doses come just eight weeks after the first dose."
Boris Johnson added: "We will also prioritise first doses for anyone eligible who has not yet come forward including the over 40s."
Earlier this week, Andy Burnham, Mayor of Greater Manchster, raised concerns about the spread of an Indian variant (B1.617.2) in Bolton, which has the highest infection rate in the country (152 cases per 100,000 people) after 437 new cases were recorded in seven days.
Yesterday The Yorkshire Post revealed 19 cases of Indian variant in Yorkshire and the Humber.
During the press conference Mr Johnson said he is "very sorry" for those living in Bolton and Blackburn where there is a surge of the Indian variant and he will not impose further restrictions on them.
But he confirmed, during question time during the press conference, that Colonel Russell Miller, commander army HQ north west, is being deployed to support local leaders in managing the response to the Indian variant on the ground.
The response will include surge testing, mobile tesing units and the army will be on the streets handing out tests.
There will be "targeted new activity" in Bolton and Blackburn to accelerate the vaccine take-up, including longer opening hours at vaccination sites.
"We are going to have to live with this new variant of the virus for some time," Mr Johnson said.
Professor Chris Whitty, England's Chief Medical Officer, said there is no doubt UK cases of the Indian variant are up, and that there's "confidence" it's "more transmissible" than existing strains.
He said that scientists are working to find out how much more transmissible it is.
It's "quite widely seeded" in places other than Bolton, but there's not a greater increase in rates among over-60s than other age groups, Prof Whitty added.
From May 17, indoor hospitality can reopen and indoor entertainment can resume, including cinemas, museums, and children’s play areas.
Up to six people or two households will be able to meet indoors and up to 30 people outdoors.
All remaining outdoor entertainment will be allowed to reopen, such as outdoor cinemas and performances.
Addressing the public, Mr Johnson said: "We won't be preventing businesses from reopening on Monday but we will be asking you do to your bit."
That includes getting the vaccine when eligible, getting free lateral flow tests and isolating when asked.
He added: "I have to level with you that this new variant could pose a serious disruption to our progress and could make it more difficult to move to step four in June.
"I must stress that we will do whatever it takes to keep the public safe."
Mr Johnson said that if the variant turned out to be much more transmissible than other variants, the country could face "hard choices".
Support The Yorkshire Post and become a subscriber today.
Your subscription will help us to continue to bring quality news to the people of Yorkshire. In return, you'll see fewer ads on site, get free access to our app and receive exclusive members-only offers. Click here to subscribe.