The rules for safety in London are the same as in New York or any big city. If you're carrying a considerable amount of cash and do not have a safe in your hotel room, it's a good idea to keep it in something like a money belt, but don't get cash out of it in public. Keep a small amount of cash for immediate purchases in your pocket or handbag.
Beyond that, use common sense. In central London, nobody will raise an eyebrow at tourists studying maps on street corners, and don't hesitate to ask for directions. However, outside of the centre, exercise general caution about the neighborhoods you walk in: if they don't look safe, take a cab. After midnight, outside of the centre, take cabs rather than waiting for a night bus. Although London has plenty of so-called "minicabs"—normal cars driven by self-employed drivers in a cab service—don't ever get into an unmarked car that pulls up offering you "cab service." Take a licensed minicab only from a cab office, or, preferably, a normal London "black cab," which you flag down on the street. Unlicensed minicab drivers have been associated with a slate of violent crimes in recent years.
If you carry a handbag, keep a firm grip on it (or even disguise it in a local shopping bag). Store only enough money in the handbag to cover casual spending. Distribute the rest of your cash and any valuables among deep front pockets, inside jacket or waistcoat pockets, and a concealed money pouch. Some pubs and bars have "Chelsea clips" under the tables where you can hang your handbag at your knee. Never leave your bag beside your chair or hanging from the back of your chair. Be careful with backpacks, as pickpockets can unzip them on the Tube, or even as you're travelling up an escalator.
Distribute your cash, credit cards, IDs, and other valuables between a deep front pocket, an inside jacket or waistcoat pocket, and a hidden money pouch. Don't reach for the money pouch once you're in public.
U.S. Department of State (.)