I've lived in London nearly 30 years and have raised my kids here. I'm in Chiswick, West London. London is a weird place in that it is, day to day, a really safe city, overwhelmingly. But some parts of London are a bit wild and woolly.
That said, my son, at 17, was beaten and mugged at knife-point for his phone in green, leafy, lovely Chiswick. Male teens are definitely a target because they have nice phones.
He's almost 23 years old now (with a bad phone!) and sailed through uni (UCL near Euston) and living on his own since 19. He's very street smart but unfortunately, this is born of experience.
He now lives in Elephant and Castle where I don't take my phone out at all once I leave the station to walk to his flat. Phone snatching is rife around there.
In all my years, in my own area, including my own son's mugging, I've witnessed the very occasional crime; we're talking once every 3-5 years. In Elephant and Castle, I see a phone snatching nearly every time I visit my son. It's a different ball game; different demographic, more populated, busier, hectic... all these things give way to opportunists. I think the more 'urban' and busier the environment, the more you take the necessary precautions, i.e. put your phone away, look out for cyclists. My son uses his phone when he's out and about, just shields it with his hands. I straight-up keep mine in my bag until I'm at a restaurant/pub/meeting point (in his area).
I do notice a lot of people don't walk around with their phones in their hands nearly as much. And I notice this because far less people walk right into me/my kids because they're so busy looking down at their phones, texting. There is an up side to crime! I'm kidding, of course.
Just keep your wits about you but don't be scared. London is definitely not a frightening city. It has its sharp edges and smooth bits like all big cities. It's gritty but not a death trap. Definitely walk around with caution but not fear. You definitely don't have to have your back up all the time.