CHATTANOOGA, Tenn.—A barren patch in the rugged hills along the Tennessee River is a sign of how Mexico has accelerated past the U.S. South in the global competition for auto investment.

The tract of cleared woodland lies alongside a factory Volkswagen AG set out to build in 2008. VW took an option on the adjacent 800 acres as a place where its Audi unit might build a North American plant someday.

But...