It may interest readers to learn that the cab rank rule so often espoused by the bar (‘Cab rank myth’) does not apply to hackney carriages either.
Both within and outside London (in England and Wales) there is no requirement for a passenger to use the first cab on the rank. It is perfectly permissible to approach a cab further down the rank because it is, for example, newer or cleaner.
Notwithstanding that, it is highly unlikely that the second cab will take you, simply because they will not want to fall out with the driver in front.
James Button, principal, James Button & Co, Matlock, Derbyshire; author of Button on Taxis: Licensing Law and Practice