A reaction set in around 1960 when simpler designs began reaching the market. And given the three or so years lead-time from sketchpad to production, stylists were probably thinking of simplification as early as 1957.
An example of incipient change might be the 1958 Ford Motor Company show car called La Galaxie. It included some jet plane detailing, but lacked tail fins and a three-tone paint-job.
There are few decent photos of La Galaxie on the internet. Some of the ones shown below had to be cleaned up, and still aren't top-drawer quality.