The worst affected areas will likely be the Highlands, in north-west Scotland and the Pennines in northern England.
Met Éireann forecaster Elizabeth Coleman said there have been some wintery showers in Cork and Kerry this morning - and more are expected in other parts of the country later.
A yellow weather warning was in place for parts of the country on Monday, with the Met Office warning that temperatures would plummet "widely below zero" leading to icy stretches on the roads and up to 2cm of snow.
The Met Office has also forecast fresh south-westerly winds for the region.
Some counties including Buxton in Derbyshire were already hit with the snowfall on Saturday, leaving fields blanketed in snow and winter service vehicles out in force to clear the roads.
The status Yellow weather warning is for all of Ireland and is valid until 10am tomorrow.
The Met Office warned up to four inches (10cm) of snow could fall in parts of Scotland today.
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There is still a chance of snow later in the week, as well as drier, colder interludes, with an increased risk of frost and fog.
The Met Office forecast for this week reveals the weather will likely be less windy with more dry weather for many.
Northumberland village Carrshield was blanketed in snow as voters braced the cold to head to their local polling stations on December 12.
Highest temperatures ranging from 2 or 3 degrees in the north to 6 or 7 degrees in the south, temperatures will be lower in any areas with lying snow.
Mr Dewhurst said: 'Anyone who is out Christmas shopping will have to wrap up warmly.
On Tuesday, cold temperatures will hit much of the United Kingdom as rain in southeast gradually clears with fog over central parts gradually lifting.
"It will be a cold start potentially to the start of Sunday morning, and it does look as though we will see some sunshine first thing".