Commuters are being warned of possible disruption, with freezing conditions and snow showers in some places.
Yellow weather warnings for snow and ice are in place for large parts of the country for the rest of the week, the Met Office said.
It is expected to be “one of the coldest weeks of this winter so far”, Met Office forecaster Craig Snell said.
Temperatures could plummet to -7C in parts of Scotland, Wales and the Midlands.
The Met Office warns of icy patches along the east coast, bringing potentially difficult driving conditions throughout Monday morning.
Kent Police has warned motorists to take care and expect sudden weather changes as it reported a series of crashes on the M20 overnight.
Up to 3cm of snow could fall inland, mainly over hills above 100m, while many will start the working week to freezing temperatures between 0C and -2C.
“It’s going to be a cold week, plenty of dry weather around, but many places will probably see some snow at some point during the week, but for a lot of us not really amounting to much at all,” Mr Snell said.
The Local Government Association said 1.5 million tonnes of salt was stockpiled ready.
“Gritters will be out treating thousands of miles of roads as the big chill bites, and councils will be keeping people up-to-date about weather forecasts, road conditions and gritting activity,” said transport spokesman councillor Martin Tett.
Mr Tett encouraged people to look out for vulnerable and elderly neighbours, who may need assistance in the cold weather.
On Monday evening through to 15:00 GMT on Tuesday, there is a chance of snow and ice for the north of England, the northwest Midlands, northern Ireland, Wales and Scotland.
“Quite a lot of the UK will see some snow as we head through Tuesday, but as it ventures into the Midlands, south-west England and eventually later in the day across south-east England, it’s just going to be a few flakes,” Mr Snell said.
“Don’t expect to build a snowman.”
The cold snap is expected to last until next weekend, with milder weather on the way the following week.