Temperatures are expected to stay well below freezing overnight and combine with wintry showers to create icy conditions, prompting the Meteorological Office to issue weather alerts.
Snowfall on Sunday evening forced airports to close their runways and left drivers struggling to get through.
Road users in the south-east were being asked by National Highways not to travel unless it was essential, with heavy snow already falling on Sunday evening and up to 10cm forecast.
Kent was being particularly hard hit, with snow severely affecting the M2, the M20 around junctions 8 and 9, the A21 and the A249, with drivers struggling to get through, the organisation said.
“We have been out gritting throughout the afternoon and we’re continuing to treat routes so we can reach the worst-affected areas and support our customers with their journeys," said Gina Oxley, the National Highways operations manager for the region.
“For anyone thinking of travelling tonight, our advice would be not to unless absolutely essential as heavy snow is expected to continue until 9am tomorrow.”
Cold weather in the UK – in pictures
Airports were also being affected, with Gatwick and Stansted having to close their runways to clear snow on Sunday.
“London Stansted Airport’s runway is currently closed to allow for snow clearance due to weather conditions and all flights are currently suspended," the Essex airport said.
“Delays to flights were experienced earlier due to de-icing of aircraft, which is a ground handler’s responsibility, and safety of aircraft and passengers is paramount.
“Passengers are advised to check with their airline for current status of their flights.”
Gatwick Airport posted on Twitter for Sunday and Monday: Snow and freeezing weather may cause disruption at the airport. Passengers are advised to check flight status with their airline — and also local travel conditions — before departing for the airport.”
“It will continue to be a very cold day, with maximum temperatures of 1ºC to 4ºC on Sunday," said Met Office meteorologist Rachel Ayers.
"And as we head into the evening, we see persistent rain, sleet or snow in the far south-east of England, which we have a yellow warning out for from 6pm on Sunday.
“We could see 2cm to 5cm [of snow], perhaps up to 10cm in some places, with Kent and Sussex most affected, with areas most exposed such as North and South Downs and higher ground going to see the more significant accumulations.
“There will be ice forming, particularly near to the coast where there is rain and sleet, and freezing temperatures that could cause some travel disruption to start Monday with that valid until 9am.”
Ms Ayers said that although cold temperatures, freezing fog and wintry showers are expected through the week, cloud cover could prevent some of the more extreme temperatures experienced in recent days.
She said there was a possibility of slightly milder conditions arriving next weekend but it was too early to be certain.
The Automobile Association has advised motorists to adjust their driving for the freezing conditions on Monday morning.
“On a frosty morning, hazards like black ice can prove lethal if you don’t adjust your speed and driving style in colder weather," said Sean Sidley of the AA.
“Always leave plenty of space behind other vehicles as stopping distances can be 10 times longer on icy roads.
“Allow extra time, as there may be delays to your normal routes, and make sure you pack winter essentials in the car such as warm, waterproof layers, a shovel, a torch, fully charged mobile phone and a flask of hot drink.
“Take heed of warning lights in your vehicle and make sure you check it regularly if you drive less in the winter.
"Flat batteries are one of the top breakdowns we attend in cold weather and can be avoided by regular car maintenance.”
The recent cold snap has led to the £25 ($31) cold weather payment to people on the lowest incomes in hundreds of affected post code districts in England and Wales.
The government payments are given to eligible households in areas where the average temperature has been recorded, or forecast to be, 0ºC or below for seven consecutive days.