The Met Office said the conditions could lead to travel disruption, especially on Monday morning.
Large areas of the country have been warned to expect severe conditions during the weekend, with snow forecast for Scotland and the south-east of England.
Temperatures are set to stay low, hitting minus 10C (14F) in isolated areas on Saturday and Sunday.
A Met Office yellow warning for snow and ice is in place for much of Scotland until 12pm on Sunday.
On Saturday, the Met Office extended ice warnings in Northern Ireland, Wales and south-western England to cover snow until 12pm on Sunday.
On Sunday, a snow and ice warning kicks in at 9am for most of London and some of south-eastern England until 9am on Monday, with a 30% chance of up to 5cm of snow.
The Met Office said the conditions could lead to travel disruption, especially on Monday morning, and a small chance of some rural communities becoming cut off along with a possibility of power cuts and mobile phone coverage being affected.
The UK Health Security Agency (UKHSA) sent out a level three cold weather alert covering England until Friday having extended the alert from Monday.
Met Office chief meteorologist Steve Willington said: “It is staying cold with daytime temperatures remaining only a few degrees above freezing in many places over the coming days and overnight temperatures dropping to minus 10C or lower in isolated spots.
“Although below average, these temperatures are not that unusual for this time of year.
“There is still a risk we could see some freezing fog in places particularly southern England, especially for Sunday and Monday mornings.
“There is also a small risk of a band of sleet or snow moving into the far South East on Sunday. If this happens it could potentially bring some disruption, especially to rush hour on Monday. A warning has been issued.”
The UKHSA is advising people to look out for friends and family who are vulnerable in the cold and to ensure they have access to warm food and drinks, adding that people should maintain indoor temperatures of at least 18C (64.4F).
Dr Agostinho Sousa, consultant in public health medicine at UKHSA, said: “Cold weather can have serious consequences for health and older people and those with heart or lung conditions can be particularly at risk.
“If you have a pre-existing medical condition, you should heat your home to a temperature that is comfortable for you.
“In rooms you mostly use such as the living room or bedroom, try to heat them to at least 18C if you can. Keep your bedroom windows closed at night. Wearing several layers of clothing will keep you warmer than one thicker layer.”
Darren Clark, severe weather resilience manager at National Highways, said gritters would be out to keep motorways and major A-roads open.
He said: “National Highways is committed to treating every road which needs to be treated – whenever it is needed.”
The RAC has reported it has been “exceptionally busy” in recent days receiving a quarter more breakdowns than is normal for this time of year.
Also, more than 3,200 warm banks, run by local authorities and charities to provide heating to those who cannot afford to heat their homes, are currently open across the UK, according to the Warm Welcome Campaign.
It said many of them are a third or even half full and offer a variety of services, from hot tea to a place to work.
Charity Save the Children said 194 of 355 councils in England and Wales were directly involved in or supporting local groups to open warm spaces this winter.
Becca Lyon, head of child poverty at Save the Children UK, said: “Families should not be in a position where they are agonising over whether to put the heating on in sub-zero temperatures.
“Parents have told us they will risk going into debt to keep their children warm.”
Councillor Richard Wenham, vice-chairman of the Local Government Association’s Resources Board, said: “Warm hubs and other similar schemes are among the many actions councils are taking to support people in the greatest need this winter, but these emergency schemes should not become the norm and are not a sustainable solution to bridge the gap between income and the current cost of living.”
It comes as people on the lowest incomes in hundreds of affected postcode districts in England and Wales are set to receive a £25 cold weather payment.
The Government payments have been triggered for eligible households in areas where the average temperature has been recorded as, or is forecast to be, 0C or below over seven consecutive days.