Not exactly the first hand experience you're looking for OP - but DH is a taxi driver (not in London). He drives a black cab so can do street pick ups but is also linked to a firm so gets pre booked appointments...the cab is rented from the firm he works for.
I can't speak to Uber but I imagine it's broadly similar to being a taxi driver in many ways.
We know drivers who make £70k + gross (after all costs, so only income tax/NI to pay). These bloke's would likely also rave about the fortune they earn for such an easy job - BUT these are the guys who work 70 hour + weeks and permanent night shifts.
Working night shifts is far, far bigger money, especially weekends. For instance, DH will earn more in 4 hours from 1am to 5am on a Saturday than he does in an entire 10 hour Tuesday day shift. Rates are higher, tips much more generous.
The flexibility to a point is great - DH never misses any important events, he just doesn't sign up for a shift. He has complete control over weeks off and hours worked etc. If we have a big holiday coming up, or in the few weeks before Xmas he can slam in a few extra shifts to up his income.
However - if you mean flexibility to do hours here and there, day shifts 9-5, regular school pick ups and weekends off and don't want any antisocial hours - if Uber is similar to taxiing he'll be lucky to scrape £25k ime. To earn decently, you need night shifts, the biggest earner being picking up the drunk clubbers at weekends.
DH really enjoys it and it works really well for us. We've found a happy medium for work/life balance. DH works three day shifts a week Tue-Thurs 6-4 and Friday and Saturday nights 4pm-4am and he takes about £40-£45k gross (after car costs, diesal etc so just tax and NI to pay). It wouldn't be anywhere near this without the two night shifts though.
One downside/consideration is that he also needs a strong stomach because he WILL have spewers if he does night shifts...i reckon DH has one a month. Some people just can't deal with that. And if you don't want to lose your entire night's earnings you need to deal with the clean up yourself there and then. DH has a kit he keeps in the boot - multiple carrier bags, cloths, antiseptic wipes, febreeze, gloves etc.
It's also easy to get sucked into chasing the money and it's something DH has to really resist. It's difficult for him to finish at 4am on a Sunday morning when it's been busy, town is still teeming with clubbers and he knows he could make an extra £250 if he works till 6! But he cuts it off to make sure he doesn't work so late he has to spend all of Sunday in bed.
Anyway, I'm rambling but hope it's of some use. Just beware of thinking it's plentiful money for nice hours only because if you could earn £47k for a standard weekday 9-5 a lot more people would be driving!