Snow is likely in northern Scotland, although temperatures will be low enough to make it a possibility anywhere in the country.
Frost and ice are also expected.
People are being urged to use their heating, despite rising energy prices, and to look out for people who are especially vulnerable.
The UK Health Security Agency (UKHSA) and Met Office have issued a level three cold weather alert between 18:00 GMT on Wednesday and 09:00 on Monday, 12 December, in most parts of England. The level also alerts social and healthcare services to take action to protect high-risk groups.
A yellow weather warning for snow is also in place for northern Scotland on Wednesday.
BBC Weather's Simon King said that, from Tuesday night and for the rest of the week, a northerly wind from the Arctic will turn things much cooler, with temperatures nationwide ranging from highs of 1C to 4C, several degrees below the average of 6C to 9C.
"We're also going to see widespread hard overnight frost with temperatures dropping below freezing, hitting -2C to -6C by the end of the week," Mr King said.
"For most parts of the UK what we'll see is sunshine - many more of us will have those cold, crisp, sunny mornings."
While snow is only forecast in north east Scotland, "the situation is that it will be cold enough for anywhere in the UK to see snow."
Higher than average temperatures throughout October and much of November, combined with rising energy bills, meant that many people have put off turning the heating on to warm their homes.
However, this could prove dangerous for vulnerable people.
The World Health Organization considers an "adequately warm home" as reaching as 21C in living rooms and 18C in bedrooms - but studies have shown that the average temperature that people will be living in if they can't afford to heat their homes is only 10C.
Age UK spokesperson Sophie Barrett urged people to put the heating on and ensure it reaches a level high enough to stay warm at home.
"We are hearing lots of very sad stories, ranging from different extremes - some are sacrificing food for heating, others are saying they will be staying in bed all day to avoid getting cold," she said.
"And some people told us they will shut appliances such as fridges off, which is dangerous from a food safety perspective, or use candles to avoid turning on the lights, which is a clear fire hazard."
She also said it is essential people look up what benefits they are entitled to to ensure they are getting the help they need for the winter ahead.
Living in such low temperatures puts a considerable strain on the body, which has to work to warm up faster.
This can lead to an increase in blood pressure and a faster heartbeat - which in turns can exacerbate the risk of a stroke or heart attack.
At low temperatures, those who already have poor heart health and the elderly become even more susceptible to serious health issues.
Dr Agostinho Sousa, consultant in Public Health Medicine at UKHSA, said: "Cold weather can have serious consequences for health, and older people and those with heart or lung conditions can be particularly at risk."
People with pre-existing medical conditions should make sure their homes are heated to a comfortable temperature, he added.
The UKHSA has also advised that wearing several layers of thinner clothing will keep you warmer than one thicker layer. Having plenty of hot food and drinks is also effective for keeping warm.
Prof Damian Bailey, from the University of South Wales, told BBC News "the evidence clearly suggests that cold is more deadly than the heat, there are a higher number of deaths caused through cold snaps than there are through the heat snaps".
The RAC has urged motorists to check their vehicles are "winter ready", with spokesperson Rod Dennis saying "many drivers might be taken aback" after a mild autumn.
A study published earlier this month suggested exposure to indoor cold not only increases risk of respiratory and circulatory illness but may also harm mental health.
Becoming unable to keep the home adequately warm leads to "statistically significant increases in the odds of reporting severe mental distress" for both those with no mental health problems and those with borderline mental health problems, the study said.
Last year, a report by the Building Research Establishment (BRE) suggested poor housing conditions in England alone cost the NHS £1.4bn - and that half of that sum (£857m) can be attributed to residents' exposure to cold in their homes.
Watch: A minute of money-saving tips with the BBC's Matt Taylor and Colletta Smith