1) Public transport is good, when it works. It often doesn't.
2) Public transport is good, until you have a job that has structured hours- which is most jobs, as evidenced by the INSANE rush hour. 20 minutes in some guys armpit feels like multiple hours.
3) You can't afford to live somewhere nice. I mean, I earned quite a bit more than the average and was spending 50-60% of my take home salary on rent. And I lived in a bad neighbourhood. (because I wanted to live alone).
4) Pubs have the same problem as the transport system, super crowded at rush times.. spills into the streets, especially in the west end/theatreland.
That said, if I had the money to live well in London, it would be a great place to live.
I'd add one more:
5) Forget going to Premier League games every weekend, it's near impossible to get a ticket if you don't know a season ticket holder!
Shitty quality of life due to absurd land costs. Of course people will say you should live like a student in shared accommodation when you have a good job aged 35. Forget it.