Except for all of the disabled ones, the wheelchair users, the ones who only wish to travel a very short distance and are unable to do so alone (the pregnant, mothers with pushchairs, the extremely inebriated, elderly, injured, etc).
The monopoly and some privileges (bus lane access, pick up fares anywhere) is granted in exchange for a lot of obligations including accepting every fare in the centre of London even at a loss (of time spent against the profit of a fare), for fixed prices, and without excluding anyone of any ability.
If Uber wish to gain the advantages that black cabs have, then they are free to accept the responsibilities and obligations that come with it. Which will of course mean changing their whole fleet to fully accessible vehicles and not allowing any driver to refuse a fare.