So I checked and found this http://www.transportforall.org.uk/news/sharp-upsurge-in-tube-lift-closures-due-to-shortages-of-trained-staff.The other day I saw a guy in an electric wheelchair get on the subway without assistance. He entered backwards so the wheelchair could sort of fall into the train. He got off without assistance also. She says she can't do that because the wheels on hers are too big.
That having been said, handicap access on public transit in Europe is significantly worse than in the US. Her blog post says she wasn't allowed on a bus because strollers were taking up the wheelchair spaces--even if she's a faker, that's a problem.
She also writes about elevators not working:
Can someone from England explain this? Does England still require elevator operators?
TL;DR - they require a member of staff present at the station in case there's a fire or emergency so that they can operate the lift in that situation. It's not that they always operate the elevator, but they wouldn't let it run without some oversight, I guess? Edit to add: for context, at most stations the lifts are hardly ever used because there are escalators or stairs instead.
And yes in general, accessibility on the Underground is shit. It has a lot of very old stations, they're underground, and they're small spaces so you can't just stick a lift wherever you like. Something like 20% of them are listed so there would be restrictions on what you can do. There are sizeable gaps between the train & the platform in many stations, there's not even much space on the train for a wheelchair on some lines (the best option I can think of would be placing it near the doors, but you'd be slightly in the way of anyone getting on and off, and Londoners don't like people in their way...). Some stations and lines have been renovated and are probably fine now, but it's a huge expense to do that and can take years.