The argument that black cabs are making is that Uber is using a taxi-meter for their fares and its illegal to have a taxi-meter installed (in London) unless you are a black cab.
(I'm making no comment about whether that rule is reasonable, I don't know why it exists other than, presumably, to deter non-black cabs from answering hails - the price needs to be agreed which should be done at booking time)
Black cab drivers are complaining that that law isn't being enforced for Uber, hence their protest. TfL have said that they don't consider using an app, having a meter installed.
At the end of the day this can only be decided by:
a) repealing the law - Uber is welcome to lobby to get that done - but they haven't.
b) bringing a test case - this is where I suspect the black cab drivers problem is. It's probably TfL who has to bring the test case. The courts will then have to decide whether an app is an "installed taxi-meter"
After (or possibly before) b, parliament can decide to clarify the law. Generally parliament doesn't act unless there's a perceived problem though - so it won't be until: 1) The courts rule that an app isn't an installed taxi-meter but parliament decides that they intended to catch the Uber case - the law will be modified to make it explicit that an app counts as a taxi-meter.
2) The courts rule that an app is an installed taxi-meter but parliament decides that that wasn't intended to be caught and clarify the law (probably after lobbying)
3) There are a series of high profile assaults/robberies/etc by Uber drivers so parliament clarifies the law so then TfL prosecutes Uber drivers.
Black-cab and mini-cab services coexist in London. I've used both and no doubt will again in the future.
Uber appears to be treading the line between a mini-cab service (which would be legal) and a black-cab service (which would be illegal). One of the great things about London is that, late at night, when you're the worse for drink, you can get into some random strangers car and be as confident as it's possible to be that that person will deliver the promised service.
There's quite a lot of (TfL) advertising warning people that "unless it's pre-booked it's a stranger's car".