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 because it uses the same line as the Eurostar, initially ran a reduced service on Sunday but by 10:30 GMT said it had reinstated its full service.
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.
Gusts of 50-55mph (80.5-88.5km/h) are likely across wide areas, with potential for gusts of 65-75mph on the southern coast of England and in South Wales, the Met Office said.
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.
Services on the Eurotunnel - which shuttles vehicles under the Channel - are running smoothly.
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 for more 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.
On Saturday some passengers were in tears on both sides of The Channel as they became stranded or saw their plans fall through.
There was a more optimistic mood at St Pancras on Sunday with one woman, who was travelling to Paris to celebrate New Year's Eve, telling the BBC she was feeling "so much better" and felt "lucky to be here today".
"[Yesterday] was a whirlwind but we all stayed calm considering," she said.
Another woman who was travelling back home to Amsterdam said Saturday had been "very, very stressful" but she was feeling "really happy that I am able to go back home".
But Lionel Buchaillot said he was waiting to try to get a new reservation after his train was cancelled on Saturday evening.
"Hopefully we will be back to France tonight. I'm reasonably optimistic because they told us yesterday they were going to have two or three more trains," he said.
He added that his son had a party in the evening so would "kill us" if they did not make it back.
It was a different story on Saturday, with a heavily pregnant woman describing how 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 the next available train was on 3 January - the day she is due back at work - with hotel and alternative travel being unaffordable.
Ms Gatier, who is 33 weeks pregnant, said: "There are no trains, no ferries, no hotels.
"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."
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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".
"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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2023-12-31 09:13:06Z
CBMiJmh0dHBzOi8vd3d3LmJiYy5jby51ay9uZXdzL3VrLTY3ODUxMDUy0gEqaHR0cHM6Ly93d3cuYmJjLmNvLnVrL25ld3MvdWstNjc4NTEwNTIuYW1w
