By Luke McGee, CNN

(CNN) -- The United Kingdom is entering its most challenging weeks since the start of the Covid-19 pandemic, a top official said Monday, as hospitals face being overrun and morgues fill up.

"We're now at the worst point of this epidemic for the UK. In the future we will have the vaccine, but the numbers at the moment are higher than they were in the previous peak — by some distance," England's chief medical officer Chris Whitty told the BBC, adding that he expects the next few weeks to be the "most dangerous time."

The country, which has already suffered more deaths as a result of the disease than any European nation and recently became the fifth nation on earth to reach the grim milestone of three million cases, is on the verge of seeing its hospitals overwhelmed.

Whitty told the BBC on Monday that there were currently more than 30,000 patients in hospital, compared to 18,000 during the first peak of the virus in the UK in April.

"We're now at a situation where in the UK as a whole, around one in 50 people is infected, and in London it's around 1 in 30," Whitty said. "There is a very high chance that if you meet someone unnecessarily, they will have Covid."

His warning comes with the country barely a week into its third national lockdown. But fears are growing that Britons are increasingly giving up on complying with the rules, as case numbers continue to surge despite the extreme measures.

Whitty stressed that minimizing contact with others will stop the situation from getting worse.

"Every single unnecessary contact any of us have is a potential link in a chain of transmission that will lead eventually to a vulnerable person," he told the BBC. "So, the absolute key is for all of us to think do we really need to have this contact?"

Whitty's intervention comes as the number of daily deaths in the UK remain very high, a point grimly illustrated by the fact that in one county in southern England, bodies are being stored at a temporary facility as morgues there are at capacity.

The temporary facility in Surrey, south of London, can hold an extra 800 bodies, on top of the 600 that can be held in morgues.

A spokesperson for the Surrey Local Resilience Forum told the UK's PA news agency: "To put some perspective on this, during the first wave, they had 700 bodies go through that (temporary) facility ... The first wave lasted approximately 12 weeks from mid-March to mid-May ... Since December 21, after just two and a half weeks, they have had 300 bodies go through it."

The UK has been ahead of the curve on approving Covid-19 vaccinations and on Monday the government is expected to outline how it will hit its target of vaccinating 13 million people by February 15.

A large part of the program will be handled by vaccination centers around the country -- the first of which opens on Monday -- and an army of volunteers who have been trained to administer the vaccine.

And even the good news that two million people have been vaccinated has been soured by reported shortages of the vaccine in some hospitals. It is unclear why the shortages are happening; the government has faced criticism for how it plans to prioritize handing out the doses it does have.

If Whitty's worst fears become reality, then the National Health Service will be under enormous strain as it attempts to cope with unprecedented hospital admissions, deal with dead bodies, vaccinate the most vulnerable citizens while also carrying out the normal procedures.

The government will be hoping that Whitty's stark warnings force citizens into complying with the measures to stop the spread of the virus.