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Election chiefs told The Times that the widespread failures to deal with high voter turnout may lead to re-runs in the next few weeks, which could be critical to the overall outcome.

The Electoral Commission said it would be conducting a “thorough review” to determine what went wrong.

An estimated 500 people were turned away in Nick Clegg’s constituency of Sheffield Hallam as a large number of students and other voters descended on the polling station at St John’s Church in the Ranmoor district of the city. Police had to remove about 100 people who refused to leave.

A further 200 people in Woodseats, a suburb five miles away, were told that they could not cast their vote. Police were called to deal with 100 angry people who refused to leave the library being used as a polling station.

About 600 people were turned away in Chester, a key marginal where Labour was defending a majority of 915.

Police dispersed 300 people who were prevented from voting in Brockley in the Lewisham-Deptford constituency, southeast London. A sit-in was also broken up in Triangle Road in Hackney, East London.

Other polling stations in Newcastle East and Sutton Coldfield stayed open after 10pm to cope with demand. Electoral Commission rules state that votes can only be counted if ballot papers have been issued by the 10pm deadline. In Liverpool some polling stations were reported to have run out of ballot papers.

John Mothersole, returning officer for Sheffield, acknowledged the shortcomings there. “We got this wrong and I would like to apologise,” he said. “We were faced with a difficult situation with the numbers of people, and a large amount of students turning up to vote without polling cards.”

Students said that part of the problem was that more than 5,000 students had been registered at a single polling station. Elizabeth Eele, 19, a physics and philosophy undergraduate at Sheffield University, said that students may have been unfairly treated because they were moved into a separate, longer queue. “[The presiding officer] said that people with children and older people should go first, but that means that students who queued for longer were not allowed to vote.”

Alice Meakin, 18, a pupil at Sheffield High School, was turned away despite turning up more than an hour before the polls closed. “At school we’ve had talks saying how it is really important that we should vote and now we can’t.”

Some voters travelled to Mr Clegg’s house in Sheffield to voice their complaints. Robin Dallman, 25, a postman from Eccleshall, in the Sheffield Hallam constituency, said: “People were angry that there were only three people in the polling station when there should have been six or seven. It was a bit of a nightmare. We went round to Nick Clegg’s house. We wanted to post our polling cards through the door, but the policeman said, ‘You can’t knock on the door.’ ”

A protester at Woodseats said he queued for two hours in the rain to be told that he could not vote. “It was utter chaos and no one seemed to know what they were doing. Everybody was understandably upset at missing out.”