Born in Catlettsburg, Kentucky on July 8, 1846. His parents, James Willets and Mary McClellan Beal Allison were both born in Susquehanna County, Pennsylvania. After their marriage, they moved to Kentucky where they were engaged in shop-keeping. At the outbreak of the Civil War, the Allisons moved to Ironton, Ohio where his father began working in the iron industry, and Henry Allison went to the local schools. He then also began working in the iron industry. In 1868 Allison moved to Hazleton, Pennsylvania, and began working in coal mining, later moving to Allentown where he joined the Allentown Rolling Mills in 1875, advancing into management as secretary and treasurer. In 1888 he was approached by the Republican Party to enter politics, which at first he declined but when others dropped out he was nominated to become mayor. During his term, streetcar service changed from horse-driven to electric powered in downtown. He left office in 1890, becoming the first President of the Livingston Club and other civic organizations. In 1893, he again became mayor, filling the unexpired term of Col Samuel Lehr, then was elected again in 1894, stepping down in 1896. His term was noted by the expansion of the Allentown business district west to 12th Street.
