Cancer patients whose screenings have been cancelled and Brits hit by the looming recession will be among the toll.
It came as ministers admitted the virus would have killed fewer if testing had started sooner.
Sir Ian Diamond, Office for National Statistics head, spoke of “indirect deaths” caused by Covid-19 on BBC1’s The Andrew Marr Show.
He said: “Changes in the prioritisation of the health service, for example, reductions in cancer screening, will lead to deaths over the next few years.
“If we have a lengthy and deep recession . . . that can lead to increased deaths as people are pushed into lengthy periods of unemployment.”
He said the high number of current deaths being seen was down to the UK having the world’s “best, most transparent and quickest recording”.
Asked if the UK was heading for the worst death toll in Europe, he told BBC One’s The Andrew Marr Show: “I wouldn’t say that at all and I would say that making international comparisons is an unbelievably difficult thing to do.
“I’m not saying that we are at the bottom of any potential league table – it’s almost impossible to calculate a league table – but I’m not prepared to say that we’re heading for the top.”
Transport Secretary Grant Shapps, meanwhile, became the first Cabinet minister to say the toll would be lower if testing had been ramped up sooner.
He said: “If we had had 100,000 test capacity before this thing started many things could be different.”
Fellow Cabinet minister Michael Gove said the Government had made “mistakes.”
He said: “When we’ve got this virus under control we can ask ourselves some deep and probing questions about lessons we can learn from how we handled this virus in its early stages.”