Flannery was born in Kentucky on 6 October 1898. He studied at the Art Institute of Chicago (IL); the Chicago Academy of Fine Art; and in London, England, at the Slade School with Frederic Victor Poole. He specialized in painting the thoroughbred horse and racing subjects. Having retired from a successful career in advertising and become a breeder of thoroughbreds on a small scale, he began painting purely for his own enjoyment. Not having any need to generate income with his art, he refused to paint commissioned scenes or portraits of particular horses. Noted racehorse owner John Hay Whitney surmounted this obstacle by sending the artist to his stud farm in Kentucky on the understanding that Whitney would buy anything Flannery produced; the Maryland Jockey Club obtained scenes of the Pimlico race track by a similar process.
Flannery exhibited a number of works at the Pennsylvania Academy of the Fine Arts in Philadelphia, among them Broodmare Barn in 1935, The Maryland Hunt in 1940, and Interior: Racing Barn in 1952. He also exhibited at the Carnegie Institute in Pittsburgh, PA; the Whitney Museum of American Art in New York City; the Corcoran Gallery of Art in Washington, DC; the Art Institute of Chicago (IL); and the Fine Arts Museum of San Francisco, Legion of Honor. The National Museum of Racing and Hall of Fame in Saratoga Springs, NY, has his Fitzsimmons Saddling, showing the trainer James ("Sunny Jim") Fitzsimmons saddling the thoroughbred racehorse Keynote while an assistant saddles another horse in the background. Studio of the Old Master, which is in the Carnegie Institute, depicts the living room of Fitzsimmons' training cottage near the back stretch of the Aqueduct racetrack in New York. Other institutions holding his work include the Metropolitan Museum of Art in New York City; the Phillips Collection in Washington, DC; the Baltimore (MD) Museum of Art; and the Toledo (OH) Museum of Art.
Flannery died in Philadelphia, PA, on 25 December 1955.