Many passengers are still stranded in London and Paris after flooding cancelled all Eurostar trains.
Eurostar has said it will run all services to London, Paris, Brussels and Amsterdam on Sunday following a day of major disruption.
New Year plans for thousands were in ruins after flooding in a tunnel under the River Thames led to the cancellation of all Eurostar services between London and Paris on Saturday.
Eurostar warned Sunday could still see delays but services have resumed.
There is also disruption on Thameslink services due to staff shortages.
Southeastern, which cancelled its high-speed services to Ebbsfleet on Saturday, as it uses the same line as the Eurostar, will run a reduced service on the route on Sunday.
And the Met Office has warned there could be disruption to domestic journeys this weekend, as wind and rain sweep across the UK. A yellow weather warning for wind is in place from 10:00 GMT to midnight for the south coast of England, south west England and south Wales.
In Scotland ScotRail has said it is expecting disruption, including speed restrictions, due to adverse weather.
The first Eurostar train left London a few minutes late at 08:10 GMT.
On Sunday morning, Eurostar said: “Flooding in the Thames tunnels has been brought under control by Network Rail High Speed.
“There will be some speed restrictions in place this morning which may lead to delays and stations are expected to be very busy.”
It said there would be no additional trains on Sunday and said customers should visit the Eurostar website formore information on compensation.
On Saturday passengers faced expensive hotel bills, significant difficulties getting to their destination or costly airfares. The Port of Dover said on Saturday that there was no remaining foot passenger availability for the day.
Richard Thorp, engineering director for HS1 which runs the track, apologised to customers saying he knew disruption to travel plans was “devastating”, but said things were looking “far more positive” on Sunday.
He told the BBC water had been cleared from both tunnels and it was now about getting as many trains and people through as possible.
An unprecedented volume of water had overwhelmed the pumping systems causing the flooding, he said.
Stories have been emerging from passengers facing difficult situations on both sides of The Channel.
A heavily pregnant woman from Norwich said she “sobbed for about an hour” after becoming stranded in Paris.
Ella Gatier, her four-year-old son Xander, and his father, were due to travel back to England after a break in Disneyland Paris.
She told the BBC on Saturday morning that the scene at Gare du Nord station was chaotic and no help was available for affected travellers.
Ms Gatier, who is 33-weeks pregnant, said the next available train was on 3 January – the day she is due back at work – with hotel and alternative travel being unaffordable.
“There are no trains, no ferries, no hotels”, she said.
“I do not have £1,200 a night to stay in Paris. I cannot even get a train or any connections into Amsterdam and back across home to England.
“In addition, I am unsure if they will even allow me to fly at this stage in pregnancy.”
Also stuck at Gare du Nord were Curt Downs, his wife Megan and their one-year-old son.
“Eurostar staff there were completely overwhelmed and couldn’t really suggest anything for us,” he told BBC News.
One staff member told them they had 4,000 passengers to assist, Mr Downs said.
He added that the family spent two hours trying to find a way back to the UK, looking for ferries, car hire and flights.
They managed to get some of the last seats on a £450 flight from Paris to Manchester, from where Mrs Downs’ mother is doing a five-hour round trip drive to get them home to Bedfordshire.
Meanwhile, at a crowded London St Pancras station, emotional travellers sat on suitcases, frantically trying to find alternative routes.
Christina David, 25, and Georgina Benyamin, 26, from Sydney, saw their train cancelled after travelling around Europe on a budget for three weeks.
They planned to “go hard” for their final stop in Paris – where they hoped to celebrate the New Year at an expensive hotel with a view of the Eiffel Tower – before flying home.
Ms Benyamin said she wanted to see Paris “light up” but was now feeling frustrated and angry.
“There were lots of people crying,” said her friend, Ms David. “We don’t know where to go, we have nowhere to stay.”
A video taken inside the flooded tunnel shows water gushing onto the tracks from a pipe attached to the tunnel’s wall.
Thames Water had earlier said a “fire control system” was likely to have caused the flooding. But HS1 said the source of the flooding will be the subject of an investigation, but at this stage it had “no evidence to suggest that the fire control system was related to the issue in any way”.
It said flooding was “being resolved” and the line would be operational, but with speed restrictions in place and delays and disruption expected.
“We understand how frustrating this has been for passengers and apologise for the inconvenience caused at such an important time of the year,” a spokesperson said.
It is the second time in 10 days there has been major disruption to Eurostar services, with a “last-minute strike” by French workers halting trains before Christmas.
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