Forecasters said the public should be aware of the potential risk of transport disruption.
Flood alerts were in place across the South and South East following heavy rain showers, with up to 30mm (1.2 inches) expected to fall in some parts by early today.
Last night the Environment Agency had more than 60 flood alerts in place, mainly in the South East and South West.
The Met Office snow and ice alert said: “Accumulations of 2-4 cm (0.8 to 1.6ins) of snow are likely above about 100m, but also to low levels locally, and more widely over the northern half of Scotland, where more than 10cm (4ins) is likely above 300 metres.
“During Tuesday the risk of snow extends into northern England, especially across the Pennines and higher parts of Cumbria. Meanwhile, a more persistent spell of sleet and snow may settle at low levels across Scotland on Tuesday night and into Wednesday morning.
“The public should be aware of the risk of disruption to transport.”
The high winds were stirred up by an extra-powerful jet stream in the Atlantic, itself triggered by plunging temperatures in the United States hitting warmer air from the south, according to forecasters.
In Brighton, two men were swept out to sea in gale-force conditions at the weekend and three people had to be rescued after being stranded in a blizzard in the Cairngorms.
Today the snow gates were closed at Braemar, Tomintoul and Cock Bridge and motorists around the country have been advised to drive with caution.
Some ferry services are also facing disruption.
Electricity has been restored to all properties left without power following last week’s storms.
At the height of the bad weather, a total of 120,000 homes in Scotland lost power and a number of lorries overturned on motorways.
Engineers battled with extreme weather to reconnect homes in the north of the country, which remained cut off over the weekend.
Scottish Hydro Electric Power Distribution said that all customers were reconnected by 10pm last night.
Many parts of the UK are on snow alert as wintry showers threaten to disrupt travel.
Frequent blasts of sleet and snow were forecast in Scotland and Northern Ireland into today and are expected to affect northern England later.