Eurostar has been criticised for its response
Eurostar is due to resume operation of its high-speed trains through the Channel Tunnel on Tuesday after three days with no service.
But passengers with pre-booked tickets are being advised not to travel unless it is essential, as the operators try to clear a backlog of journeys.
The carrier will be concentrating on passengers who booked on the weekend.
The series of train breakdowns which caused the chaos have been blamed on condensation on electrical parts.
Eurotunnel, which operates the car-carrying trains through the Channel Tunnel, experienced severe delays on Sunday because of very heavy traffic.
It said its Folkestone terminal was at saturation point for several hours. Shuttle services are expected to continue operating on Tuesday, but Eurotunnel has warned customers without bookings to stay away.
The head of Eurostar's operations Nicolas Petrovic told the Associated Press news agency that the shutdown had affected 40,000 people.
Eurostar launched an inquiry into the breakdowns.
Transport Minister Sadiq Khan said he had asked for the review to report not to the Eurostar board, but to him and to shareholders.
And French Transport Minister Dominique Bussereau said his ministry would launch a probe into "what happened, how it happened and how to avoid such events in the future".
The BBC's Greg Woods said Eurostar will struggle to clear the backlog before Christmas Day, even if there are no more breakdowns.
The firm said snow shields used to protect the trains' electrics had worked for the past 15 years, but the recent cold snap in France had been "unprecedented" in their experience.
Richard Scott, BBC business reporter
Eurostar sells itself as being the easy way to get to the continent, claiming it's doubled the number of people travelling from London to Paris and Brussels. So the sight of angry and exhausted passengers waiting fruitlessly in stations is the last thing it wants.
The company has commissioned an independent review of what's gone wrong, admitting it could have handled things better - especially where communication was concerned. The company accounts for three quarters of the traffic between London and Paris, and it's possible that an air of complacency crept in - and that the problems with snow getting into trains' electrics could have been predicted.
But the tests carried out by Eurostar on the new modifications seem to have been a success. And just like Heathrow Terminal 5 and its calamitous opening, past problems tend to fade in the public consciousness. So it's likely there'll be little lasting damage to Eurostar. As long as this doesn't happen again, of course.
Eurotunnel's earlier passenger backlog - which had brought delays of at least two hours - appeared to have been sparked by hundreds of passengers who turned up on Monday after postponing their travel plans over the weekend because of the bad weather.
Those - including disappointed Eurostar customers - who turned up hoping to buy a ticket on the day were turned away.
The company said drivers with pre-booked tickets between 2030 GMT on Monday and 0900 GMT on Tuesday should turn up as normal.
But firm spokesman John Keefe urged passengers travelling after 0900 GMT on Tuesday to consult its website or telephone information line to help plan their journey accordingly.
Mr Keefe said there had been almost 7,000 cars booked to travel on Monday.
He said: "In order to help the traffic flow we closed the check-in and then reopened it, letting in traffic in batches. We've got the terminal into free-flow again."
And the firm has asked passengers who had planned to travel over the next few days to change their bookings if possible.
Their priority was to get displaced people back home, it said.
Eurostar said it hoped to run two-thirds of its trains on Tuesday but its limited service is likely to start later in the morning than usual.
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