I think there'll always be a place for Black Cabs, but they'll have to fight for it and accept changes to remain competitive. And that place will inevitably shrink a little in any case.
Black Cab drivers are professional drivers. Yes, we've all been given grief by the gobby ones and some of them think they own the road, but they know the roads inside out and they're highly regulated. The cab has a number and therefore you know who your driver is.
I've used Uber literally a couple of times, the App is brilliant and they're cheaper than Black cabs. But both times it was clear to me that my (inevitably Asian) driver was not the guy in the pic on the App. It's not a problem for me because I'm a hairy-arsed bloke, but for a woman? (And yes, I should have reported it but didn't).
Why mention the Asian thing? Well it's probably relevant because I should imagine there are members of the same household/extended family who run that car. You might even consider that as relevant in the current Covid climate of course.
I think Black Cabs have things they can learn from Uber, but they need to get their fingers out. Yes, they will always be at a disadvantage because they're more tightly regulated, but that aspect will also be an asset of course, especially for those that don't mind paying the extra.
I guess you might liken it to Royal Mail/Parcelforce dealing with the threat from Hermes/DPD/UPS etc. I sell all my old bits and pieces on eBay, and in terms of trackability, cost and insuring sent items, Royal Mail always seem to lose hands down when I search online for the best delivery option. However, I bought something from an Italian supplier recently, and there was suddenly proper tracking all the way until there was a knock on my door...by my postie. It seems they've suddenly joined the 21st Century. About bloody time!
It's basically sink or swim, isn't it?