At an abandoned home with yellowing newspapers on its front stoop, Paul Palmer peeks through a mail slot to find letters and leaves carpeting the entryway. The house next door has a dead plant chained to its porch, which is covered in faded utility bills.

Mr. Palmer investigates abandoned homes for a living. But his turf isn't a poverty-stricken corner of this financial capital. It's the Mayfair district, home to wealthy financiers, celebrities, the U.S. Embassy -- and a few squatters.