There are fears that snow and ice could cause travel disruption later as temperatures plunge.
Snow has fallen in parts of the UK, as the country braces for a week-long cold spell.
South-east England will see a mix of snow, sleet and rain during Monday, BBC Weather said.
The Met Office has warned of ice and snow across southern England and Wales later, which could cause travel disruption.
Flooding could also continue, mainly in central England, the Environment Agency said.
The Met Office has issued a yellow warning for ice for southern England and southern Wales, from 15:00 GMT Monday until 03:00 on Tuesday. It warns that ice and small amounts of snow “could lead to slippery surfaces in new places” and potentially slower journeys.
BBC Weather presenter Stav Danaos said north-east England will see light rain, light sleet and snow over hills, while south-east England will experience wintry showers and “a light dusting of snow”, even in lower areas.
The Met Office said “a mix of sleet and snow showers” will move in from the east, with temperatures reaching “near zero”.
“Given these wintry showers, and also wet surfaces after recent wet weather, some icy patches are likely on untreated surfaces,” the forecaster added.
Temperatures are expected to drop heavily on Monday night. The Met Office said parts of England and Wales could reach -4C, while northern Scotland could see temperatures as low as -7C.
On Monday, the UK Health Security Agency (UKHSA) and the Met Office issued an amber cold weather warning for the South West, South East, West Midlands, East Midlands and North Westparts of England until 12:00 on Friday.
An amber alert means that the impact of cold weather is likely to be felt across the whole health service for an extended period of time.
Dr Agostinho Sousa, from UKHSA, stressed the importance of checking on those who could be vulnerable.
For older people in particular, cold weather can increase the risk of heart attacks, strokes, pneumonia and worsening arthritis. There can also be an increase in accidents at home, due to a loss of strength and dexterity in the hands.
Mayor of London Sadiq Khan has activated an emergency severe weather plan to ensure councils across the capital open additional emergency accommodation for people sleeping rough. A similar scheme has been activated in Reading, offering a bed to anyone who is sleeping rough.
A yellow warning for ice was also issued for parts of Northern Ireland, with the Met Office warning of potentially difficult travelling conditions caused by icy patches on some untreated roads, pavements and cycle paths.
Police have urged road users to exercise caution.
“Stick to main, gritted roads when possible. Slow down, and increase your braking distance from the vehicle in front,” a Police Service of Northern Ireland spokesperson said.
Nick Powell, from the AA, said anyone travelling in extreme weather conditions should check the condition of tyres – including the spare – and fully de-ice the car, which includes clearing snow from windows, lights and the roof “so you can see and be seen”.
“It’s also worth having winter essentials in the car such as warm, waterproof layers, a shovel, a torch, and a flask of hot drink”, he said, while phones should also be fully charged.
“The cold snap is also likely to impact vehicle breakdown levels, with faults such as flat batteries and frozen windscreen wipers,” he said.
How is the weather affecting you? Get in touch.
Meanwhile, the Environment Agency (EA) has said “significant river flooding impacts” are expected on Monday in parts of the Midlands, Lincolnshire and on the River Thames.
EA flood duty manager Katharine Smith urged people not to drive through flood water, and to follow advice of local emergency services on the roads.
“Flood water is often deeper than it looks and just 30cm (11in) of flowing water is enough to float your car,” she said.
The warning follows a week of heavy rainfall last week, some of which came as part of Storm Henk.
The EA warned that more than 1,800 properties have already flooded, and more could be affected over the next week as river levels rise.
Thousands of homes and businesses are at risk of flooding in Surrey due to rising water levels of the River Thames, while an MP in Oxford has said the wait for a flood alleviation scheme is “frustrating” after parts of the city were submerged after Storm Henk last week.
Flooding minister Robbie Moore, making a statement on Storm Henk, told MPs on Monday 2,000 properties have been flooded due to Storm Henk.
He said: “There is now an improving picture across the country, but as we enter a dry spell existing flood warnings always still remain in place and we will continue to monitor the situation very closely.”
He added the government would invest a £5.2bn in flood defence schemes over the next six years.
Shadow environment secretary Steve Reed called for more to be done, highlighting that the EA had found that more than 4,000 flood defences were considered to be in a poor or very poor condition last year.
Labour leader Sir Keir Starmer has said the government’s response to flooding is not “good enough”, and said his party would have established a task force earlier in the year to tackle the problem.