Arctic air is expected to bring snow, cold winds and icy conditions to parts of the UK from Sunday evening onwards.
A Met Office yellow warning began at 18:00 GMT in parts of north-east Scotland. The warning extends south into England on Monday and Tuesday.
Met Office meteorologist Craig Snell said up to 20cm (4in) of snow could fall on high ground, with up to 10cm possible in low-lying regions.
The weather could disrupt travel and other day-to-day activities.
Places covered by Sunday evening’s warning – issued earlier the same day – include Aberdeen and Dundee, the Highlands, Orkney and Shetland.
The warning for Monday says the affected area will encompass more of Scotland and a corridor of north-east England that extends to Yorkshire.
For Tuesday, the warning area grows further to cover Strathclyde, more of Yorkshire and the Humber, and the East Midlands. Frequent snow is expected.
The Met Office said snow could cause delays on roads in these places, as well as rail and plane cancellations. It also warned of the risk of slips and falls on icy surfaces.
There was “slight chance” that rural communities could be cut off, it said, adding that cuts to power and phone services were possible.
BBC forecaster Sarah Keith-Lucas said that Tuesday night could be the coldest of the week and one of the coldest of the year, with lows of -10C (14F) possible in Scotland.
She added that it was “not too unusual” to get these “cold wintry blasts” even after entering meteorological spring, due to the effects of air from the Arctic.
But the forecast for the middle of the week is less certain, with the arrival of warmer air from the Atlantic expected to create a “battleground” over the UK, she said.
She said this could bring snow to more southerly parts of England, or some much-needed rain following a dry six-week period.
Earlier alerts issued by the UK Health Security Agency also remain in place. These warn that all of England is likely to experience cold weather for much of next week.
The agency said this could have a “serious impact” on the health of those who are vulnerable to cold weather, and urged people to check on relatives.
It advised over-65s, or those with pre-existing medical conditions, to heat their homes to at least 18C.