BBC Weather predicts "a very slippery start for just about everywhere" on Sunday.
While there could be snow flurries for much of the UK, as much as 8cm is forecast for south-east and eastern England, then north-east England later.
Wintry weather caused widespread travel problems on Friday and Saturday.
Heathrow Airport will be hit by a third day of cancellations and delays, with a planned 20% reduction to the flight schedule on Sunday because of forecast snow and low visibility.
The Met Office has issued a yellow "be aware" ice warning covering almost the whole of England, all of Wales and Northern Ireland, and the south-west corner of Scotland.
The warning says: "The public should be aware of the risk of localised difficult travel conditions due either to existing snow cover, or to icy surfaces."
There are also yellow snow warnings for eastern parts of England and Scotland, stretching across the Midlands into north Wales.
Wales could face the coldest weather on Sunday, with forecasts suggesting temperatures could fall as far as -5C overnight and fail to get above zero all day.
In other developments:
- Four climbers were killed in an avalanche on a mountain in Glencoe, in the Highlands, on Saturday afternoon
- In County Down, about 150 homes are still without power. Engineers had worked through Friday night and Saturday morning after more than 2,000 homes had lost power owing to heavy snow and high winds
- The M48 Severn Bridge is covered in 4in of ice and will remain closed in both directions until 09:00 GMT on Sunday at the earliest
The heavier snow is expected to reach north-east England by Sunday evening, then push into eastern Scotland, with the possibility of wintry weather continuing in those parts into Monday.