They discovered they could buy sensitive information on someone for £8/person. This included not only their credit card details, secuirty codes, addresses, date of birth, passwords and transaction history but also recordings of customer call centre conversations with the client! The investigator brought about eight to 'test out'. She then went to the house of one of the customers she'd brought the infornation of and showed him it all only to discover he was actually already disputing some of the charges on his account!
The banks they mentioned included Alliance & Leicester, Halifax, NatWest and possibly Barclays (my friend who watched the programme can't confirm that for certain though).
The Times also ran an article on Call Centre Fraud last Sunday (Credit Data Stolen At Indian Call Centres : The Times Online; Sunday 1st October 2006)
EDIT: The Register also reported the problem and the Channel 4 investigation yesterday (Indian Data Theft "Exposed" : The Register; Thursday 5th October 2006)