Eurostar's Richard Brown said people needing to travel urgently will be prioritised
Eurotunnel has closed its shuttle car service to new arrivals, after saying its terminal was at "saturation point".
Day trip bookings are not being honoured, with Eurotunnel citing a "massive backlog" partly due to three days of problems on the Eurostar.
But passengers with pre-booked tickets are no longer facing any delays.
Meanwhile Eurostar has announced it hopes to resume a limited service on Tuesday, saying: "We will do our best to get everybody home by Christmas."
Eurostar services were suspended on Monday for a third day. The firm's head of operations Nicolas Petrovic told the Associated Press news agency that the shutdown had affected 40,000 people.
Eurostar launched an inquiry into the train breakdowns, thought to have been caused by condensation forming on electrical systems as the train entered the warm tunnel.
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.
Transport Minister Sadiq Khan said he had asked for the review to report not to the Eurostar board but to him and to shareholders.
Eurostar said snow shields used to protect the electrics had worked for the past 15 years, but the recent cold snap in France had been "unprecedented" in their experience.
Contractors were busy fitting modifications to the trains in an attempt to make them less prone to breakdown.
Services on Tuesday would depend the result of further tests on modified trains on Monday afternoon, said Eurostar chief executive Richard Brown.
But he said the firm hoped to run two-thirds of the normal service - amounting to about 26,000 seats - with more services planned on Wednesday and Thursday.
Only customers who hold tickets that were originally for travel on Saturday 19 or Sunday 20 December will be able to travel on Tuesday.
Customers holding tickets for Monday and Tuesday will be able to travel on Wednesday, and those with tickets for Wednesday and Thursday will be eligible for travel on Thursday.
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.
If your journey is not essential, do not travel
A full refund will be offered to those whose journeys have been affected
Passengers whose journeys were severely disrupted on Friday or Saturday will also be given £150 compensation, out-of-pocket expenses, and a free return ticket
Those who had bookings for Saturday, Sunday or Monday can claim "reasonable" out-of-pocket expenses such as hotel, transport and meal costs
All updates will be posted on the
and given out to news outlets
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