Drivers are being urged to take extra care in icy conditions to avoid an unwelcome slide into the new year.
Overnight temperatures dropped to a low of minus 7.3C (18.9F) in Wiltshire, and frost and ice are expected to last into next week, the Met Office said.
Areas in the south of England and parts of Scotland will see some showers which, combined with single-digit temperatures, will make roads treacherous.
Police forces in areas hit by ice and snow urged motorists to travel only if necessary as emergency crews attended several collisions on New Year’s Eve.
Derbyshire Police reported a five-car collision just before 7.30am on New Year’s Day.
Humberside Police also warned drivers to be careful in a light-hearted post on social media, showing an officer casually sliding down an icy road in Scunthorpe.
“Sliding into 2021 like … the A18 Mortal Ash Hill is closed into Scunthorpe this morning due to treacherous conditions. Please take care, very icy out here,” the force tweeted.
Traffic Scotland said “visibility is vital” and warned motorists in Scotland to clear all snow and ice from their vehicle windows before starting journeys.
Met Office meteorologist Alex Burkill said the UK will see “harsh frosts” throughout the first week of January.
“Obviously it’s very cold and it’s going to stay cold through this week,” he said.
“Whilst there will be some wintry hazards around, it’s not really until the end of the week until we see any significant snow.
“Tonight we’ll see temperatures dropping again and that will bring a fairly significant risk of ice patches, particularly in northern parts of Scotland.
“I expect it won’t take very long at all for the risk of icy patches to start forming an hour or so after sunset.
Mr Burkill added that temperatures in the South East may dip just below freezing on Friday night and parts of Scotland could face lows of minus 7C (19.4F).
He said a high of 4C (39.2F) is expected in London on New Year’s Day, with temperatures dropping to around freezing overnight.
Further north, in Manchester temperatures are expected to reach a high of 5C (41F) but plummet to a low of minus 4C (24.8F) overnight.
No weather warnings are currently in place after the Met’s ice warning, covering the majority of Devon and Cornwall, expired at 9am on January 1.
RAC Breakdown spokesman Simon Williams said: “We strongly urge drivers to exercise caution on New Year’s Day, with a high risk of both snow and ice on the roads.
“The message for those who have to drive is to adjust their speed according to the conditions and leave extra stopping distance so 2021 doesn’t begin with an unwelcome bump and an insurance claim.
“Snow and ice are by far the toughest driving conditions, so if they can be avoided that’s probably the best policy.”
It came after the final day of 2020 brought bitterly cold conditions for much of the UK and more snow – with a low of minus 7.5C (18.5F) recorded in Topcliffe, near Thirsk in North Yorkshire.