The men - aged between 26 and 47 - were arrested Friday and accused of installing a device on the bank's computer system made it possible to carry out the cyber theft.
Police said cash, jewels and thousands of credit cards were found in related searches of addresses in the greater London area.
Police said Michael Victor Harper and Lewis James Murphy were charged with conspiracy to commit fraud by false representation. Darius Bolder faces that charge and a charge of conspiracy to steal, while Tony Colston-Hayter is facing one count of conspiracy to steal.
The four will appear at London's Westminster Magistrates Court on Saturday.