During rush hour on a crisp March evening, Adrian Fox whooshed through London’s narrow streets, dodging double-decker buses and overtaking several of the capital’s familiar black cabs. Then, suddenly, he took a sharp turn to the left. “Don’t worry, it’s a shortcut,” Mr. Fox said, pedaling through a labyrinth of small, dark streets.

Mr. Fox is a rickshaw driver, one of a growing number swarming around central London every evening to pick up tourists and theatergoers for a ride around town.

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