Car hire can work but it's often more expensive, needs to be booked ahead of time and they also refuse trips that aren't profitable for them.
Uber openen up travel for many people in London, especially at night. It's quite easy to find a lot of places in London that aren't serviced by Night Buses, there are plenty of places that are 15-30 min walk from a tube station, and the Night Tube is only available on select lines and only during the weekend.
I don't know if they still do, but many nightclubs would have some arrangement where there was someone from a minicab company in the lobby or just outside. You'd say "W3", he'd say "That'll be £20, in 10 minutes time" and you could wait in the warm until the car arrived. (Of course, if you knew a number for a different company, you could phone them yourself.)
The drivers were pretty clueless -- even a fairly large inner-London station name like "Ealing Broadway" would frequently be misunderstood, and they'd want to slowly type a postcode into a satnav -- but otherwise they were OK.
Even in the center good luck hailing one at 2am on a Saturday night.
I’ve been refused rides form the Shard, Bank, and Canary Wharf plto Lancaster gate, both of which are in Zone 1.
If I want to hail a cab from my home I have to walk 10 min to Paddington and I live essentially across from Hyde Park which makes me considerably more central than most people in London.
For my friends who live in Hammersmith, Fulham, Acton and Chiswick they might as well be living in Belgium as far as black cabs are concerned they don’t go there and they won’t take you there.