A retired British businessman accused of conspiring to sell missile components to Iran is to be extradited to the US within 10 days after a last-ditch plea to the European court of human rights was rejected.

Christopher Tappin, 64, from Orpington, Kent, denies unlawfully attempting to export batteries for Hawk air defence missiles and says he was the victim of entrapment in a sting organised by US government agents.

His long legal battle through the UK courts to block his removal ended in failure last month when high court judges Lord Justice Hooper and Mr Justice Cranston refused to allow him to take his case to the supreme court.

Refusing permission, Mr Justice Cranston said time would now begin to run out for his removal to the US.

Now an application to the human rights judges for Rule 39 relief to stop Tappin's removal pending a hearing of his case in Europe has been rejected by the court in Strasbourg, his lawyer, Karen Todner, said on Monday.

Todner said: "We truly hope that the UK government will see this case as an example of the gross injustice to British citizens by the UK/USA extradition treaty and, as they declared in opposition, they will now act quickly to make the necessary amendments."

An independent review of the UK's extradition arrangements by Sir Scott Baker last year found that the treaty between the US and the UK was both balanced and fair. But critics claim it is one-sided and the latest development will increase pressure on the government to ignore the review's findings and renegotiate the treaty.

It comes ahead of a meeting between David Cameron and Barack Obama at the White House next month.

The high court judges rejected Tappin's challenge to a decision by district judge John Zani at City of Westminster magistrates court last February that extradition could go ahead. They said the entrapment argument was unsustainable and extradition would not be oppressive or a breach of human rights.

The judges added that the charges Tappin faces were extraditable offences, and they had to accept the extradition request was made in good faith.

Tappin, a former director of Brooklands International Freight Services, Surrey, said he had become trapped in a nightmare and believed he was exporting batteries for the car industry in the Netherlands.

Mr Justice Cranston said the allegation against Tappin was that he had participated in a conspiracy with another UK citizen, Robert Gibson, who operated an export business in Cyprus, and an American, Robert Caldwell. Others involved have not been named.

An investigation was launched by the US department of homeland security, using a shell company, Mercury Global Enterprises, staffed by customs agents.

In 2007 an agent charged Tappin and the other alleged conspirators with criminal offences. Gibson agreed – unknown to Tappin – to co-operate with the US authorities and told them he was buying technology for a customer in Tehran.

Gibson pleaded guilty in a Texas court and was given a two-year jail sentence in February 2007. A jury convicted Caldwell in June 2007 of aiding and abetting the illegal export of Hawk missile batteries, and he was sentenced to 20 months' imprisonment.

Mr Justice Cranston said a warrant was issued in February 2007 for Tappin's arrest, but nothing happened until almost three years later when the US government submitted an extradition request.

Edward Fitzgerald QC, for Tapping, said extradition would be oppressive under Article 8 of the European convention on human rights because of the delay and the fact that Tappin was caring for his sick wife.

Rejecting the submission, the judges said "serious offending" was alleged against Tappin. According to the extradition request, Tappin had been involved in the conspiracy since April 2006 and was due to receive half of the profits.A number of other Britons are fighting extradition to the US, including 23-year-old student Richard O'Dwyer, who is accused of breaking US copyright laws.

The mother of computer hacker Gary McKinnon said he was "unable to control the terror that consumes his every waking moment" as he fights extradition.

Janis Sharp said the treatment of her son, who admits hacking into military computers but claims he was looking for evidence of UFOs, was barbaric as she marked 10 years since his first arrest. She called for the prime minister to raise the issue with President Obama next month.