Wilson E. Schmidt, 54, a professor of economics at Virginia Polytechnic Institute and State University and a former professor at George Washington University, died of cardiac arrest Tuesday at George Washington University Hospital.

He was hospitalized for injuries he received in a fire at the Cosmos Club Sunday night. Fire officials said the blaze apparently started from a cigarette.

Dr. Schmidt was President Reagan's nominee to be the U.S. executive director of the World Bank. The nomination was announced in June and was awaiting Senate confirmation.

Dr. Schmidt, who was born in Madison, Wis., graduated from the University of Maryland in 1947. He earned a master's degree at the University of Pittsburgh and a doctorate in economics at the University of Virginia.

In 1950, he joined the faculty of George Washington University as an instructor and rose to the rank of professor. In 1966, he moved to VPI at Blacksburg and was head of its economics department from 1966 until 1977. He waqs on leave from the faculty at the time of his death.

From 1970 to 1972, Dr. Schmidt was a deputy assistant secretary of the Treasury. He also had been a visiting professor at the School of Advanced International Studies at Johns Hopkins University in Baltimore and at the Johns Hopkins Center in Bologna, Italy.

His publications included a book, "International Economics," which he wrote with J. N. Behrman and which was published in 1957. He was a member of the American Economic Association, the National Association of Business Economists, the Mount Pelerin Society and the Cosmos Club.

Dr. Schmidt's survivors include his wife, the former Eleanor Butt Parker, whom he married in 1950, of Blacksburg, and three sons, Eric Emerson, Carl Wilson, and Orrin Edward Schmidt.