They were among the first people to be extradited under a new extradition agreement, under which the US no longer had to show any evidence of guilt to extradite suspects, just a charge in the USA. Their lawyers fought this through various courts, saying they couldn't be guaranteed a fair trial in the US any more, and there wasn't even any examination of evidence in the UK under the new agreement.
As Wikipedia summarises:
The Three were arrested in Britain on 23 April 2004 and extradition proceedings commenced in June of that year. <7> In September a judge ruled that extradition could proceed.
Extradition from the UK to the US is governed by the controversial Extradition Act 2003.
On 20 February 2006 the Natwest Three's appeal against the extradition was rejected by the High Court, and on 21 June 2006, the House of Lords threw out the appeal;<8> on 27 June 2006 the three lost an appeal to the European Court of Human Rights. There were then rumours in the UK press that the British government would support their case but this was rejected by Attorney General Lord Goldsmith on July 7, 2006.<9>
http://en.wikipedia.org/wiki/NatWest_ThreeTheir case has generated criticism of extradition laws that mean the US is not required to provide "prima facie" or solid evidence of wrongdoing to extradite a UK citizen.
Britain must still provide the US with evidence of "probable cause" if it wishes to extradite someone from America.
The affair is being seen as a test case in the UK of the government's Extradition Act 2003 - which was developed in the wake of the 11 September attacks in 2001.
http://news.bbc.co.uk/1/hi/business/5103730.stm