More than 2,000 people were stuck after four Eurostar trains broke down, while roads and air travel were also disrupted by the cold snap.
It is thought the trains failed as they left the cold air in northern France and entered the warmer air inside the tunnel, Eurostar spokesman Bram Smets said.
People were stuck in France and Folkestone, while hundreds gathered at London's St Pancras terminal.
The tunnel closure added to travel problems caused by heavy snow in Kent and police have brought in Operation Stack - which allows lorries to park along sections of the M20 - to ease congestion near the terminal.
First passengers arrived into St Pancras at about 8am and were greeted with a round of applause from the crowds waiting for friends and family.
But many complained about the conditions while trapped in the tunnel overnight.
One passenger, Edward Carey, from London, said he had boarded his train at 7.15pm last night in Paris.
He said: "We were stopped outside the tunnel and they said that we would be late but then we broke down completely inside.
"It was awful and the fumigation was handled very badly."
He said passengers on his train were then put on another train normally used for car passengers before being taken to Ebbsfleet where they waited for more than three hours for more passengers to be taken on board.
Lee Godfrey, who was travelling back from Disneyland Paris with his family, said they were left without light, air-conditioning, food or water as they spent the night in a stranded train.
Passengers opened emergency doors by themselves and set off down the tunnels to find refuge on another train, he said.
Mr Godfrey told BBC Radio 4's Today programme: "We were without power. We ran out of water, we ran out of food and there was very very poor communication from the staff.
"We lost air-conditioning when we lost the power. We had to open the emergency doors ourselves. The evacuation procedure we followed was one that we set down ourselves.
"We have had children asleep on the floor, they have been sick. It has been a complete nightmare.
"It was very scary for the children and elderly people. We had a lady in a wheelchair and a lady who was seven months pregnant and we had asthma attacks, people were fainting on the train. It was just pandemonium. People were very, very panicky."
Mr Godfrey said his family left Disneyland Paris at around 7.30pm French time yesterday and after 14 hours had only got as far as Folkestone, having first been taken back to Calais this morning where they were given food and drinks.
Another passenger, Philippe Serna from Lincolnshire, said the worst thing about the journey was it got a bit "stuffy" when the air conditioning failed.
He said: "I got on about 5.30 in Paris and we got into the tunnel about half way through I think when we stopped.
"We were stuck there for about four hours and then they announced there would be another train coming to meet us.
"We got out and walked down the tunnel only a short distance, and Kent Fire Brigade were there, and they put us on a shuttle which took us to Folkestone where we got back on another Eurostar which brought us here.
"The journey was very long but it was fine to be honest just boring, frustrating and a bit stuffy."
Dozens of people were waiting at St Pancras to pick up friends and relatives.
David Taylor, of Brentwood, Essex, said his daughter had been due in at 10:30pm last night and he spent the night waiting for her.
"My daughter is a student and was working in Paris and ran for her train at about 8pm only to find it cancelled," he said.
"She got the next one but was stuck in the tunnel for six hours. It's pretty desperate now with no food and no drink."
Mr Smets said the Eurostar trains affected were all going from Paris to London.
"It's really snowing in northern France. The weather conditions are very bad," he said.
Mr Smets said "very limited services" were expected to resume today and tomorrow but passengers were advised not to travel this weekend if their journey was not essential.
A spokesman for Eurostar later said two trains had been moved back to Calais but would be taken to St Pancras International later today.
He added that the passengers were currently at Folkestone and were being put on buses to be taken to the station.
He said: "We have done all we have can to get people to their destination as quickly as possible
"Obviously we understand people's frustration."
Kent Police issued an urgent warning for drivers to stay away from the Folkestone and Dover areas of the M20.
The force brought in Operation Stack at 5.30am which will allow more than 2,300 lorries to be parked on sections of the motorway.
Traffic is also queuing on the A20 into Dover after sailings were stopped due to severe weather in France which left vehicles unable to leave Calais.
The Port of Dover later said the French authorities were allowing tourist traffic only to disembark and ships were now sailing from Dover.
However, freight was still restricted.
A Port of Dover spokesman said: "The port is still full and traffic continues to queue on the A20 into Dover.
"Customers intending to sail to Calais are advised not to set out for the port without checking with their operator first."
Disruption caused by snow and ice left dozens of passengers stuck on a plane at London's Heathrow Airport for more than six hours overnight.
British Airways flight 215 to Boston, Massachusetts, was due to take off at 4pm yesterday.
Passengers finally boarded at 7pm but found themselves still waiting to take off at 1am today.
A British Airways (BA) spokesman said the airline, along with all others operating out of the south of England, had been affected by the severe weather.
Budget airline easyJet said it expected continued disruption to flights today, with passengers advised to check the airline's website.
More snow is expected across the UK today and overnight.
Helen Rossington, forecaster with MeteoGroup UK, the weather arm of the Press Association, said light snow showers were expected in eastern coastal areas.
A band of snow over Scotland will move south, reaching northern England today.
The North East and Yorkshire could see between 10 and 15cms of snow on high ground, she said.
Snow will reach southern England tonight with eastern areas likely to see between two and 5cms.
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