The miners' union was broken by 1921, after President Warren G. Harding put the entire state of West Virginia under martial law and sent the army to the coalfields to defend the companies against their employees. By then, hundreds of miners had been killed, thousands arrested and jailed. It was not until 1935, under Franklin D. Roosevelt's New Deal, that union organizing was legally protected in the United States.
According to John Sayles' official web-site, this picture "has become a staple in the teaching of U.S. Labor History."
In reality, the identity of the Baldwin-Felts spy in the community (who was indeed the character identified in the film) was not revealed until the trial which followed the battle.
The nickname "Wobblies" refers to The Industrial Workers of the World (IWW) union which is an industrial global union that was established in 1905.