Heavy snow and blizzards have forced the almost complete closure of the international airport serving Iceland since Saturday.
The Foreign Office warned: "There is a severe weather warning across the whole of Iceland and many roads are closed.
"Police are advising people to stay at home. High winds and dangerous icy road conditions are causing major flight disruption and affecting movement to/from Keflavik international airport due to road closures."
Besides being shut to flights, Keflavik has been cut off from the capital, Reykjavik, as snow blocked the 30-mile highway link, Road 41. Icelandair is reported to have flown two shuttle flights between Keflavik and the city airport in Rekjavik to move crew.
Dozens of flights to and from the UK have been cancelled, with their passengers spending nights at the airport.
The relative of elderly travellers from the US who were stranded after their onward flight was cancelled told The Independent: "Growing mood of anger amid 1,000-plus passengers stranded at KEF - especially Americans - that while yes, there was bad weather, Icelandair's response has been woefully incompetent."
The national airline said: "Our team are working extremely hard to rebook all customers who have been affected by the disruptions. Please bear with us as we anticipate that it may still take some time to accommodate everyone.
"We will be working hard to run as many flights as we possibly can. However, we expect there may be some continued disruptions to flights due to the events of the past few days. Should there be any, customers will be notified through the contact information provided in their booking."
Icelandair has chartered in extra aircraft to clear the backlog.
Some passengers may finally be able to leave in the next 24 hours; Britain's biggest budget airline, easyJet, said it will operate extra flights on Thursday to clear the backlog.
Passengers with easyJet who were trying to leave on Tuesday were told: "We have booked hotel accommodation for you if required, but we have since been made aware many of the access roads to the available hotels are shut due to the heavy snowfall."
On Tuesday, the Icelandic Met Office issued severe weather warnings for the whole country, and the highway agency warned: "Many roads have already been closed."
By Tuesday evening buses were able to take many of the passengers into the capital, where the Red Cross opened shelters.
Flight departures are resuming, with a stranded British Airways jet expected to take off for London Heathrow on Wednesday afternoon.
Gisli Olafsson, MP for Iceland's Pirate Party (formed in 2012 and campaigning for direct democracy), has become a commentator on the issue. He tweeted: "Blizzard conditions for days are not that common in Iceland.
"They were caused by heavy snowfall (30+ cm) overnight on Saturday, followed by cold weather and strong winds (18-25 m/s).
"This means the main problem is snow that the wind is blowing, not new snow."
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