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 or a neck pouch, 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 directions. However, outside of the center, exercise general caution about the neighborhoods you walk in: if they don't look safe, take a cab. After midnight, outside of the center, take cabs rather than waiting for a night bus. Although London has plenty of so-called "mini-cabs" -- 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." Only take a licensed minicab from a cab office, or, preferably, a normal London "black cab," which you flag down on the street. Unlicensed mini-cab drivers have been associated with a slate of violent crimes in recent years.
If you carry a purse, choose one with a zipper and a thick strap that you can drape across your body; adjust the length so that the purse sits in front of you at or above hip level. Store only enough money in the purse to cover casual spending. Distribute the rest of your cash and any valuables among deep front pockets, inside jacket or vest pockets, and a concealed money pouch.