Richard Shenkman is taking his information from the 1997 book What Americans Know about Politics and why it Matters, and has corrected the statement in the 2010 version of his book.
The 1997 book says 91% stated "The U.S. used A-bomb against Japan" in a 1990 survey. (page 87, table 2.7)
and also says 49% "correctly" answered a survey item "only nation to use nuclear bomb" in a 1986 survey (page 88, table 2.7); and separately (page 84, table 2.5) the book says 49% for "only nation to use nuclear weapons" in a 1986 survey.
According to the May/June 1986 Deadline, Nuclear Times article "What We Know About Arms Race Is More Than a Matter of Opinion" by Jay Rosen:
To test public knowledge, the Center [for War, Peace and the New Media] designed two questions that Roper included in a recent 42-item opinion survey. In the first, Roper's 1,993 respondents were given four statements about the use of nuclear weapons and asked to choose the correct one. Forty-nine percent knew that the U.S. "is the only nation to have ever used nuclear weapons against another nation in time of war." Eleven percent believed we never dropped the bomb, 10 percent thought the U.S. and the Soviets both had used nuclear weapons, and another 13 percent said that other nations had employed nuclear weapons during a war. Seventeen percent said they didn't know.
57% responded that the United States was the first country to develop a nuclear weapon according to a 2015 survey.
According to the Quebec Agreement:
Whereas it is vital to our common safety in the present War to bring the TUBE ALLOYS project to fruition at the earliest moment; and whereas this may be more speedily achieved if all available British and American brains and resources are pooled; and whereas owing to war conditions it would be an improvident use of war resources to duplicate plants on a large scale on both sides of the Atlantic and therefore a far greater expense has fallen upon the United States;
It is agreed between us
First, that we will never use this agency against each other.
Secondly, that we will not use it against third parties without each other's consent.
The USA and UK, in accordance with the above earlier agreement, 4 July 1945 jointly agreed to use the atomic bomb against Japan as recorded in the Minutes of a Meeting of the Combined Policy Committee:
Use of Weapon Against Third Parties.
FIELD MARSHAL WILSON stated that the British Government concurred in the use of the T. A. weapon against Japan.