I made many run and gun in London, on none of them i had problem. You should be aware of putting your tripod middle of a private area or private estate. It sounds good but sometimes in london its hard to understand. You think you're in a kind of park, but it could be a private garden of a "Lord" Don't show your professional side to public, keep things simple then you'll not have any problem. If you see a police officer lookin at you just have a smoking break
But you can still find some police officers asking permission from you, they mostly don't arrest, just will warn you. And you'have to stop it immidately, at least till they go
As UK goverment thinks that everyone with a camera is a potential terorist, this site also may be helpful. There should be kind of legal form here which makes you half legal shooting in London.
PS: Don't ever try to shoot or at least try to open your tripod in a public transport like "TUBE" for example. That will be the worst thing you can do in England, IMO.
The only problem I've had filming in public in London was when I was filming in a park which was next to a prison. The security officers / wardens were very civil and gave us some time to wrap up filming.