It comes as supplies of the vaccine are set to be restricted next month, something which had prompted discussion over whether the roadmap would be delayed.
However, the prime minister, speaking at a Downing Street press conference on Thursday, insisted: “Our progress along the road to freedom continues unchecked – we remain on track to reclaim the things we love, to see our families and friends again, to return to our local pubs, our gyms and sports facilities and of course our shops.”
News of the reduced supply emerged on Wednesday evening after NHS England told health leaders to expect a significant shortfall.
Watch: 'I'm getting my jab tomorrow,' Boris Johnson says
A delay in deliveries from India and the need to retest a batch of 1.7 million doses is behind the expected shortfall in April, the government has said. Johnson said this retesting is "part of our vigorous safety programme".
The PM said there will be "slightly fewer" vaccines in April than in March, "but that is still more than we received in February".
He added next month's supply "will still enable us to hit the targets we have set", including over-50s and clinically vulnerable people being offered a first dose by 15 April, as well as second doses being available to 12 million people over the month.
Johnson also maintained every adult will have been offered a first dose by the end of July.
“So there is no change to the next steps of the road map."
The next full easing of lockdown is set to take place on 12 April. In this stage, the government plans to allow the likes of shops, hairdressers, nail salons, gyms, outdoor attractions, libraries and community centres to reopen.
Pubs and restaurants would also open outdoors – with no curfew and no requirement for alcohol to be accompanied by a "substantial meal", as was the case in the winter.
In the third stage, on 17 May at the earliest, two households or groups of up to six people will be allowed to mix indoors, and limited crowds will be allowed at sporting events.
Pubs and restaurants will open indoors, as well as entertainment venues such as cinemas and theatres. Hotels would also reopen, as well as sports stadiums subject to capacity limits.
On 21 June at the earliest, the government aims to have removed all restrictions on social contact.
Watch: How England is leaving lockdown