Oswald was born as the son of a mason in Giessen. After elementary and secondary school in Giessen, he attended a vocational school for salesmen. During World War II he was a soldier of the German army.
In 1945, he joined the Social Democratic Party of Germany. After local offices in Giessen, he was elected to the parliament of Hesse in 1954. From 1962 to 1969, he served as minister in the government of Hesse, first as minister of economic affairs, later as minister of finance. Osswald was elected prime minister of Hesse in 1969, after the demission of Georg August Zinn. His government had an absolute majority in the beginning. From 1970 on, Osswald headed a coalition government of SPD and FDP. After a financial scandal involving the public bank Helaba, Osswald resigned as prime minister of Hesse in October 1976. He retired from all political offices in 1977.
Osswald died during a hiking vacation in Schwangau in 1996.