The dramatic falls in infections were achieved primarily because of the tough restrictions – rather than the vaccination programme, the prime minister said.
“The bulk of the work in reducing the disease has been done by the lockdown,” Mr Johnson said.
“So, as we unlock, the result will inevitably be that we will see more infection, that sadly we will see more hospitalisations and deaths – and people have just got to understand that.”
However, the prime minister insisted the key future dates for lifting curbs – on 17 May and 21 June – would go ahead as planned, on the current data.
“At the moment, I can’t see any reason for us to change the roadmap, to deviate from the targets that we’ve set ourselves,” he said.
But, Mr Johnson added: “People need to understand that the success of the vaccine rollout is great – but it’s not just the vaccine rollout that’s causing the reduction in deaths and infection.
“Overwhelmingly, that’s been delivered by the lockdown and that we need to continue to be cautious and be sensible.”
He argued Nigel Boardman would be given “pretty much carte blanche to ask anybody whatever he needs” to get to the bottom of the controversy.
“I would like it to be done quickly, but wanted him to have the maximum possible access, so we can all understand exactly what’s happened”, he insisted.
Pressed on why it was not “a full independent inquiry”, Mr Johnson replied: “I’ve got every confidence in Nigel.”
And, asked what he made of his predecessor, David Cameron, lobbying on behalf of Greensill Capital, he said: “That’s a matter for Nigel Boardman’s report.”
“People have got questions that they need to satisfy themselves about – including me – about how this supply chain finance stuff is meant to work.
“I don’t think it is going on at present anywhere in government, but we need to understand exactly what the intention was, how it came about, and that is exactly what Nigel Boardman is going to do.”
Mr Johnson’s warning about future deaths came after a member of the joint committee of vaccination and immunisation warned there could be another 50,000.
Because of vaccinations, “if control of the virus is lost, then the damage it can do will be relatively restricted,” Professor Jeremy Brown told BBC Radio 4.
“But when I say relatively restricted, what I mean is that a big third wave could still end up with 30,000 to 50,000 deaths, potentially, if it was a similar sort of size to the previous waves that we’ve had.” he warned.