Further snow and ice in many parts of the UK could cause disruption later, the Met Office has warned.
Yellow warnings to "be aware" of frozen conditions are in place as temperatures continue to stay low, particularly in central and western areas.
Rail routes between London and East Anglia, parts of which are expected to see more snow, could be hit by delays because of speed restrictions.
Forecasters said the cold spell would continue until Friday.
The BBC Weather Centre predicted further snow in eastern areas, including Lincolnshire and East Anglia overnight into Tuesday, and in parts of South East England during the morning. Snow has also been falling overnight in some northern counties of England.
However, it said a more widespread problem would be ice, with temperatures widely falling below freezing - and possibly going as low as -8C in rural parts of south-west England and south-west Scotland in the early hours.
The day is expected to be drier than Monday, but some showers falling as sleet and snow will make icy conditions worse as surfaces freeze behind them, forecasters said.
Snow blowersIce could cause particular disruption to commuters in western Scotland and Northern Ireland.
On Monday, snow in large swathes of England and parts of Scotland - particularly Aberdeenshire - led to a number of problems, including school closures and travel delays.
Carol Kirkwood on where the snow and ice is due
Rail operator Greater Anglia issued a series of service alterations and cancellations for services between East Anglia and London Liverpool Street. It said speed restrictions were expected to remain on some routes on Tuesday.
Steve Crosthwaite, of the Highways Agency, said motorists should check the latest travel advice before they set out, check their vehicle was prepared for the conditions and take warm clothing and food.
He told BBC News: "We are using salt to treat the road network and have ploughs and snow blowers on standby if necessary. Our traffic officers are working around the clock to monitor our road network, deal with any incidents and keep traffic moving."
Tasmin Jeff-Johnson, from the RAC, urged motorists to take sensible shoes and a warm coat on any snow-affected journey. She also advised drivers to take a shovel and pieces of old carpet to provide traction in case of breakdown.
Continue reading the main storyThree Met Office yellow warnings - advising people to "be aware" of severe weather - were put in place for much of the UK on Monday afternoon, to run to about 09:00 GMT on Tuesday.
Eddy Carroll, chief forecaster for the Met Office, warned some snow into Tuesday could cause disruption, adding: "With some very low temperatures over the next few nights we also expect ice in many places and people should be prepared for travel problems."
The Met Office has also issued a cold weather alert, warning of a 90% probability of severe cold weather or icy conditions until Friday in parts of England.
The level three alert - one below a national emergency - warns the weather could increase health risks to vulnerable patients and disrupt services, and notifies the authorities to take action.
The cold spell is being caused by an abrupt jump in temperatures high in the stratosphere, which can bring snow, forecasters explained.
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