Storm Eowyn has been “pretty exceptional” and was “probably the strongest storm” to hit the UK in at least 10 years, the Met Office has said.
It was the most intense in “more like 20 or 30 years” for some parts of the country, forecasters added.
Parts of Ireland saw the highest windspeeds since records began after they reached 114mph in Mace Head, Co Galway, on Friday.
A gust of 100mph was recorded at Drumalbin in South Lanarkshire in Scotland that day.
Kacper Dudek, 20, died after a tree fell on his car at Feddyglass, Raphoe in Co Donegal, early on Friday and Irish police are investigating the incident.
More than a million people in the UK were without power, and there was significant travel disruption across the UK and Ireland.
The storm eased slightly overnight but those carrying out recovery work in Scotland will still face some “fairly difficult” conditions, the Met Office said.
A fallen tree which crashed through the wall of Phoenix Park and on to Blackhorse Avenue in Dublin. (Brian Lawless/PA)
A yellow warning for strong winds is in place for much of Scotland until 3pm on Saturday, where gusts could reach 50-60mph inland, 60-70mph on exposed coasts and hills and possibly 70-80mph in the Northern Isles.
Yellow snow and ice warnings are in place until 10am in Northern Ireland and 11am in parts of Scotland.
A yellow ice warning covers the East, South East and South West of England, East and West Midlands, and parts of Wales until 10am.
Saturday will be bright and sunny for large parts of the UK but a new weather front from the south west is set to bring wet and windy weather as the weekend continues, with a chance of local flooding.
Meteorologist Tom Morgan said: “The winds have still been strong nonetheless overnight, so it’s definitely not the calm after the storm today.
A fallen tree on Tullydraw Road near Dungannon. (Oliver McVeigh/PA)
“It’s certainly a calmer day, but there is still going to be strong winds around, particularly across Scotland today, but for most of the UK, it’s certainly a much brighter and less windy picture, at least for most of Saturday.
“Thousands of homes (are) still without power this morning, thousands of trees came down onto power cables, and a lot of disruption to transport will continue, probably through the next few days, and I think that there’s still going to be some fairly difficult conditions for recovery operations.”
On Saturday morning Network Rail Scotland said nearly 400 “incidents of damage” have been found including more than 120 reports of fallen trees following Storm Eowyn.
Signalling systems, overhead wires, stations, boundary fencing, level crossings and train depots were also damaged, it added.
Sunday brings a new set of weather warnings.
A roof blown off during strong winds rests on some bungalows in Amble, Northumberland, in the North East (Owen Humphries/PA)
South East, South West and North West England, as well as Wales, south-western parts of Scotland, and Northern Ireland, are covered by a yellow wind warning until 3pm on Sunday.
South to south-westerly winds will be accompanied by heavy rain and widely bring 50 to 60mph winds, the Met Office said.
Some exposed coasts and hills may feel gusts up to 70mph.
A yellow warning for heavy rain is in place from 8am on Sunday to 6am on Monday, bringing a chance of local flooding for parts of the UK.
Heavy rain is forecast in central and southern England, and most of Wales, on Sunday morning followed by heavy thundery showers.
Much of those areas will see 10-20mm of rainfall and 30-50mm is expected over higher ground.
A further heavy spell that evening could mean as much as 80mm of rain falls.
“Given recent heavy rain, this extra rainfall could lead to some local surface water and river flooding”, the Met Office said.
ScotRail previously said all of its services across Scotland would remain suspended until midday on Saturday at the earliest.
In Northern Ireland 189,000 homes and businesses remained without power on Saturday morning, NIE Networks said.
NIE Networks said around 214,000 homes and businesses remained without power in Northern Ireland.
The Scottish Government said 106,000 properties were without power in Scotland late on Friday evening.
More than 1,100 flights were cancelled on Friday, with Dublin, Edinburgh, Heathrow and Glasgow airports the worst affected.