Born in Richfield, Sevier County, on 12 September 1924, Howard Nielson attended Richfield High School and graduated in 1942. He served in the United States Army Air Force from 1943 to 1946, attaining the rank of sergeant. He attended the University of Utah and graduated with a B.S. degree in 1947. Nielson then graduated from the University of Oregon at Eugene with an M.S degree in 1949. He went on to Stanford University, where he earned an M.B.A. in 1956 and a Ph.D in 1958. In 1957 he joined the faculty of Brigham Young University, where he taught statistics and business.

Nielson also involved himself in public service by participating in the political process. He was elected to the Utah House of Representatives in 1967. He served there until 1974, including a term as speaker from 1973 to 1974. A conservative, he ran for the U.S. Senate in 1974. He believed that government spending was the cause of inflation. Nielson worked as Associate Commissioner of Higher Education from 1976 to 1978 and served as a delegate to the Utah Republican party state conventions from 1960 to 1982. He was elected to the U.S. House of Representatives from the newly created Third Congressional District in 1982 and served four terms until 1990. He then retired from public office to devote more time to his family and church. He married Julia Adams on 12 June 1948 and they have seven children.