"Despite what the media (and for crying out loud, the US President), says, London is actually a remarkably safe city."

I'm not sure what the media or president is saying, but looking at the data I would not call it "remarkably safe". It sounds like the London violent crime rate is 28 per 1000 while New York and Los Angeles are about 700 per 100,000. I wouldn't let my kids ride public transport all over those cities until they were at least teens (the laws might not allow it anyways).