Met Office warnings for strong winds and heavy rain are in effect for all four nations.
Storm Jocelyn will spread across most of the UK on Tuesday, a day after Storm Isha wreaked havoc and left at least two people dead.
Rainfall is expected in western Scotland, north-west England, the Yorkshire Dales and north-west Wales, with flooding possible.
Winds will gust at 40-50mph (65-80km/h) across the UK by late afternoon.
All ScotRail trains will be suspended from 19:00 GMT and will not restart on Wednesday morning.
The train provider said each route would need to undergo a safety inspection before being able to operate due to the risk of trees and other debris falling onto the track.
The north and west of Scotland are in store for the strongest of the winds, expected to reach 80mph (130km/h).
TransPennine Express, which operates rail services between northern England and Scotland, is “strongly urging” customers going to or from Edinburgh to not travel after 15:00.
Avanti West Coast’s last London-Glasgow service will depart at 15:30 and is expected to be “extremely busy”. The provider has advised passengers not to travel north of Preston after 15:30, adding that services to and from Scotland will be cancelled until at least noon on Wednesday.
In Northern Ireland, about 7,000 customers were still without electricity as of Tuesday morning, NIE Networks said.
Elsewhere in the UK, Electricity North West said more than 50,000 homes had suffered a loss of power, with 2,000 yet to be restored.
Jocelyn is the 10th named storm to hit the UK since September.
The Met Office has issued the following severe weather warnings:
- An amber warning for wind for the fringes of western and northern Scotland from 18:00 on Tuesday till 08:00 on Wednesday
- A yellow warning for wind across the whole of Scotland, Northern Ireland and northern Wales and north-west England
- A yellow warning for wind for south Wales, the Midlands and north-east England from 12:00 on Tuesday until 15:00 on Wednesday
- A yellow warning for rain across western Scotland from 07:00 until 18:00 on Tuesday
- A yellow warning for rain across parts of north-west England from 11:00 until 19:00 on Tuesday
- A yellow warning for rain across parts of Wales from 12:30 until 19:00 on Tuesday
With most of the UK under a yellow warning for wind, the Met Office warns there could be possible disruption to travel and there could be some damage to buildings.
Motorists are being urged to reconsider making journeys where the weather is expected to be the worst, including western and southern Scotland and north-west England.
“With so much heavy rainfall and debris on the roads, driving conditions will be very challenging,” RAC spokeswoman Alice Simpson said.
She said visibility will be “severely reduced” due to the spray from lorries and other large vehicles, while the amount of water on the roads will increase stopping distances.
“We urge drivers to consider postponing their journeys in these areas if at all possible. Those that do need to drive should try to avoid exposed coastal routes where strong winds will make driving much more difficult,” she added.
But Jocelyn will not be as powerful as Monday’s storm – which left thousands of homes without power and battered parts of the UK with gusts of up to 99mph.
Storm Isha’s winds also disrupted the journeys of many road, rail and air travellers.
An 84-year-old man died after the car he was in, crashed into a tree near Falkirk; while a man in his 60s was killed when a tree fell on his car in Limavady, County Londonderry.
Winds will only slowly ease on Wednesday – which will be a much brighter day, with a scattering of showers.
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