There is a possibility of up to 20cm (8in) of snow on the ground in areas higher than 300m (984ft) and 10cm (4in) in areas higher than 200m (656ft), the Met Office said.
There is also a “chance” that snow could hit lower levels and cause road disruption.
The Met Office said wintry conditions could coincide with rush hour on Tuesday, leading to disruption on some transport routes across central parts of the UK.
National Highways has issued an amber severe weather alert for snow in the North East and North West regions of the country from Monday at 19:00 until Tuesday at 10:00.
Darren Clark, National Highways Severe Weather Resilience Manager, advised drivers: “Keep your distance and reduce your speed, because even in conditions that seem normal, and where the snow is not settling, it can be slippery if ice patches have formed, or where fresh grit has not been worked into the carriageway.”
“Drivers should plan their journeys, monitor weather reports and pack a snow kit of blankets, food, water and a shovel.”
The UK health security agency (UKHSA) has said there is an increased risk to vulnerable people, warning the weather could impact the whole health service.
It has issued an amber cold weather health alert for the East Midlands, West Midlands, North East, North West and Yorkshire and the Humber.
“It is vital to check in on vulnerable friends, family and neighbours to ensure they are well prepared for the onset of cold weather. Particularly if they are elderly or otherwise at increased risk,” Dr Agostinho Sousa, Head of Extreme Events and Health Protection at UKHSA, said.
