It is believed a huge snow cloud now sweeping across the Atlantic will bring heavy snowfall and freezing temperatures along with it.
Snow is then forecast on Tuesday in the South and Midlands including near London, and from Wednesday in the North.
Met Office spokesman Richard Miles said: "There will be strong winds across the whole of Wales".
Isolated showers and clear spells over the eastern half of the country at first on Sunday night, but cloudy and misty conditions with outbreaks of showery rain in the west will spread to most areas overnight.
"The snow risk will continue into Wednesday with many places seeing accumulations of a few centimetres".
Forecasters warn that "showers will fall as snow over the hills" and it will be "cold with lighter winds".
Vodafone puts Huawei rollout in core networks on hold
Carriers are becoming wary of expanding their ties to Huawei amid a USA -led crackdown on the largest telecom-equipment supplier. Huawei will reveal the world's first 5G smartphone with a foldable display at Mobile World Congress next month.
It was forecast to stay cold for the rest of the week, with the potential for disruptive snow in central and southern areas Tuesday into Wednesday. It's the coldest it has been across Britain this winter, for sure.
We've already had a brisk start to 2019 and the outlook for the coming week looks just as cold.
After this cold windy day on Saturday, Met Éireann said temperatures will dip to freezing overnight and into Sunday morning with temperatures of 0 to 3 C, with some frost and icy patches.
"10C is certainly possible each night from Sunday for most of the week over lying snow in Scotland".
We could see some snow nearly anywhere on Tuesday. England could see -6C.
"Tuesday sees a snow risk over the southern half of the United Kingdom, with a possible covering on the Downs and Chilterns, and sleet in London".
Snow at the Kent and East Sussex Railway, Tenterden last February.