Black cabs have no obligation to pick up fares and will regularly refuse to take trips that they feel are inconvenient or they think will likely result in a return journey without a fare, even after you're in the cab. Want to go south and it's near the end of the driver's shift and he lives north? Back to waiting for you.
This is in comparison to the NYC system where once you're in the cab, they're obligated to take you where you want to go.
Legislation that obligates the driver to drive up to 6 miles, or for 1 hour, providing the driver has no reasonable excuse.
Later challenged by Hunt v Morgan 1948, in which it was clarified that if the taxi is on a taxi rank they have no right of refusal, and if the taxi is stopped they have little right of refusal. If the taxi is in motion then the driver may not have seen you, it may not be safe to stop, etc and the driver may have a reasonable excuse.
In nearly all cases, if a driver has stopped for you and you are travelling less than 6 miles, the driver has no reasonable excuse and is obligated.
Other parts of the legislation oblige the driver to help those who need such assistance (accessibility stuff).
The law stipulates a Hackney Carriage, once hailed, must accept any fare within the Metropolitan Police Area. If refused one should report it to the Public Carriage Office. It might not sound like it is worth it, but the PCO takes trade regulation very seriously and it will have implications for the driver, especially after repeat complaints.
This is in comparison to the NYC system where once you're in the cab, they're obligated to take you where you want to go.