Residents are advised there is an increased risk of travel delays and accidents from trips and falls.
But forecasters have warned people to expect heavier snow to arrive on Thursday and Friday.
The national forecaster says it will remain valid until 6pm on Saturday.
Cloud, rain and snow are expected to clear to give clear skies across Yorkshire.
The Met Office has issued a new warning for snow and ice this week.
The warning covers the period between noon tomorrow and 11am on Wednesday.
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Further snow showers are expected Tuesday evening (29 January) and overnight into Wednesday morning (30 January).
A spokesman for the Met Office said: "After a frosty start, Tuesday will be cold and start dry".
"More concerning as that moves to the east because it will turn very icy, so I think for many places towards the end of the night there will be very icy conditions across the UK".
A yellow warning for snow and ice is also in... In the South East we could see lows of -3C (27F) while in rural parts of the North it will drop below -5C (23F).
The Met Office is also warning that areas could be cut off by snow and that Thursday's snowfall poses the biggest risk.
Weather warnings are in place from Thursday 3pm until Friday 12pm.
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Regardless of where you live, much of the country is expecting to feel what could be record-setting lows this week. In more persistent bands in western and northern NY state, several feet of snow could pile up in some communities.
"It is easy to forget how traumatic a cold snap can be, it is important to follow the advice and keep warm, eat well and avoid going out in the worst conditions".
Up to four inches is expected to fall on higher ground and some parts are already seeing motorists struggling to get on with daily life.
A covering of 1-2cm is more likely at lower levels.
Police Scotland urged drivers to "exercise extra caution" and check for updates online.
In addition high winds could affect power supply across the country.
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