Cockney School was a pejorative label used by Blackwood's Magazine in the early 1800s for a circle of London writers associated with Leigh Hunt, and by extension for poets such as John Keats who were seen as urban, middle- or lower-middle-class, and not part of the literary establishment. It was never an actual school or movement; the phrase was meant as ridicule, not neutral description.
Example
From a narrow London lane,
he writes of rain on chimney pots,
and critics sneer, calling his verse
Cockney School poetry.