I have included some links below to the enotes study guide section on this novel which will give you more information, but basically, the Atkinson family is shown to be one of the families that make up the history of the fens that the narrator, Tom Crick, is so desperate to explore in order to help him understand his own present sufferings.
As the novel progresses and we delve deeper into Fen history, the Atkinson family is one of those families that is focused on and we see their various tragedies, triumphs and successes played before us. Alfred and George Atkinson became very successful businessmen and were men of some consequence, founding the Atkinson Water Transport Company and building the New Atkinson Brewery. Alfred became mayor of Gildsey. George's grandson, Earnest Atkinson, develops a new recipe for ale which he begins to manufacture in 1906 and wins him fame and success. He becomes enamoured with his daughter after the brewery is destroyed in a fire in 1911, and Dick Crick is born as a result. Earnest is the grandfather of the narrator of this story, Tom Crick.