The mercury will steadily fall in the coming days before a chilling dip by Wednesday.
Day-time temperatures by then will be as low as 5C but the windchill factor will make it feel more like freezing.
Overnight temperatures will be even colder.
An easterly wind is driving the temperatures down – rivalling even Reykjavik (5C), experts said.
The November average for England and Wales is 10C.
Some heavy showers are also likely turn into snow across central and eastern areas, including Lincs and the Chilterns.
Met Office forecaster Steven Keates said: "Things are going to get colder as we enter the week.
"Temperatures by Wednesday will be down to 5C but the windchill factor will make it feel closer to freezing.
"It's going to feel pretty raw. It'll be the type of wind that blows through you rather than round you.
"It's going to be quite a shock to the system."
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The expected snowfall is unlikely to accumulate or cause any major travel disruption, he added.
But the cold weather is likely to remain all week and into next weekend, with a risk of further snow showers.
Mr Keates added: "It won't feel particularly pleasant out there. It's typical cold near-Winter weather with overnight frosts and ice."
Britain was hit by the worst snowstorms in 30 years last Winter.
Temperatures were below average for each month from January through to March.
The Beast from the East and Storm Emma crippled the nation for days.
Roads were blocked, train services axed and flights grounded.
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