The French rail network was sabotaged by three coordinated arson attacks on Friday, affecting travel to and from the UK, Belgium, and across the north, east and west of France.
A fourth attack on a line running south was foiled, French rail company SNCF has said.
Some 800,000 customers will be affected with disruption expected all weekend, the rail firm says.
The disruption coincides with travelling to see the opening ceremony of the Paris 2024 Olympic Games, and comes as millions embark on summer getaways.
Two of those waiting in queues for the Eurostar at London's St Pancras station on Friday were the parents of Team GB rugby sevens star Ellie Boatman, who said they were prepared to wait "for however long".
Asked whether the delays might get in the way of their excitement, they both said "no".
"We'll just have to have another glass of prosecco," they said.
Other passengers facing long waits at St Pancras showed similar optimism.
Kate Fisher, 37, was planning to travel from London to Paris with five friends to watch the games.
"We knew this is absolutely the worst time to go to Paris because of the Olympics, so we're prepared for it to take longer," she said.
"When I saw the delay on the departure board, I was like 'whatever, as long as we're there'," she added.
Another passenger, Kate Heath, 54, said she had looked into travelling to Paris by plane instead, but she was unable to find a flight that would arrive in time for the opening ceremony.
"We'll just take it in our stride," she said.
Despite the disruption, some Eurostar services from London arrived in Paris, with delays of up to three hours, according to travellers.
Ian Ralph, 48, told the PA news agency he felt "panic" and "stressed" when he realised his train to Paris was cancelled on Friday ahead of Olympics events his family have tickets for, including the women's football on Sunday and beach volleyball on Monday.
Speaking at St Pancras, the researcher from Southampton said he managed to book him and his family onto a train at 6pm - around three hours later than his original booking.
Sir Keir Starmer changed his travel plans and took a flight to Paris due to the disruption on the Eurostar network.
He is due to attend a leaders reception hosted by French President Emmanuel Macron before going to the Olympic opening ceremony along the River Seine.
Eurostar’s Chief Commercial Officer François Le Doze said the majority of passengers hoping to leave London for Paris on Friday will still be able to do so, but with delays of around 90 minutes.
Trains will have to run slower along certain in France after the network was attacked and damaged, he said.
Extra staff at St Pancras International have helped keep things calm in London and that overall there is still a general mood of excitement, he added.
Eurostar said it expected the disruption to last until Monday morning and encouraged customers to "postpone their trip if possible".
The company said affected passengers can cancel or refund their tickets, or modify their journey free of charge.
A spokesperson for the company said: "Due to co-ordinated acts of malice in France, affecting the high-speed line between Paris and Lille, all high-speed trains going to and coming from Paris are being diverted.
The UK Department for Transport said it is "in close contact" with Eurostar and French authorities regarding the ongoing disruption.
It has advised anyone planning on travelling to check ahead before leaving and to follow advice from the Foreign Office and Eurostar.
The majority of Team GB athletes have been arriving at the Games via Eurostar, but the British Olympic Association confirmed that only two athletes were scheduled to arrive on Friday and had been subject to only minor delays.
SNCF said the situation should last "at least all weekend" but that teams were already on site carrying out checks and beginning repairs.
A spokesperson for Eurotunnel Le Shuttle, which transports vehicles from Folkestone to Calais in northern France, said its services had not been affected by the incident.
Tips to make travelling smooth as possible, whether you’re going by road, plane, train or ferry.
France's interior minister vows the attackers will be arrested fast, but does not say who was to blame.
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