U.K. Prime Minister David Cameron called for a reboot of the cozy relationship between British politicians, police and media in response to the mounting scandal over reporting tactics at News Corp.'s News of the World—just as the arrest of Mr. Cameron's former media strategist underscored how tightly wound together those worlds are in Britain.

A day after News Corp. tried to contain the cascading scandal by saying it would close the 168-year-old News of the World, Mr. Cameron's former chief communications strategist, Andy...