The company said the backlog of passengers had now been cleared but a restricted service was in operation.
Domestic rail services across the UK have been severely delayed, with buses replacing trains in many areas.
British Airways said the majority of its flights were operating again after it cancelled all its domestic and European flights from the main London airport, Heathrow, on Monday.
Other airlines, including EasyJet and Ryanair, have cancelled dozens of flights and told travellers to expect delays.
Although Frankfurt airport re-opened on Tuesday morning, passengers there still faced cancellations and delays, with 140 flights cut from the schedule.
And Berlin's main international airport, Tegel, had to be closed for an hour because of freezing rain.
In France, Belgium and the Netherlands, airport operators cleared runways after thick snowfall caused major disruption to flight patterns.
Road traffic in the UK was severely disrupted, including the south-east where roads were gridlocked late on Monday after numerous crashes in heavy snowfall.
Early on Wednesday morning, at least two people were killed and dozens injured when a coach overturned on icy roads in Cornwall, in south-west England.
Major roads elsewhere in Europe were blocked after some regions had snowfall of up to 50cm (20in).
The Eurostar crisis prompted the French transport ministry to order an investigation into the shutdown.
Meanwhile Eurotunnel - which carries vehicles under the Channel between England and France - cancelled day trip bookings on Tuesday and warned that heavy traffic levels were expected through the day.
In Poland, police appealed for people to help if they came across homeless or drunk people lying outside, as temperatures dropped towards -20C in some areas.
Most of the 79 people who froze to death in the country since the wintry conditions began were homeless, police said.
Cold-related deaths were also reported in France, where two homeless people died.
Treacherous travel conditions also caused havoc for rail services.
Fifty people were injured when a train hit a buffer in the Croatian city of Zagreb, while 36 were injured when a passenger train derailed in Paris after a car slipped on ice and knocked concrete on to the tracks.
Temperatures are forecast to rise later in the week.