When Charlie Isaacs turns on a radio, it starts playing immediately. Radios of that era had vacuum tubes, not transistors. Vacuum tubes required several seconds to "warm up" in order for the radio to play.
The women are shown with pink foam hair rollers. Doubt those were available in the 40s. More like the late 50s or early to mid 60s. In the 40s they would likely have used kid (leather wrapped around wire) rollers or some kind of twist to set their hair themselves. Or they would have used hot curlers and then pin the curls and wrap their heads with a silk scarf.
Frank includes the Shah of Iran as a potentially dangerous possessor of atomic bombs which made no sense since Iran was a neutral country and an ally of the west after the war until the radical Ayatollahs overthrew the government.
At the beginning of the episode the radio announcer is listing off war deaths and several Lance Corporals, the Marine Corps did not officially use this rank until 1958.
The ribbons that Colonel Cox has on his uniform include the World War II Victory Medal, the Army of Occupation Medal (for Germany/Japan occupation after the war), the National Service Medal (first available ten years later for service in the Korean War), the Korean Service Medal, and the United Nations Korea Medal.
Frank Winter claims to an MP that the Vice President doesn't know what the Manhattan Project is and thinks it's a leaky tunnel on the IRT. While that would later be true during the brief Vice Presidency of Harry Truman, the episode is set in July 1943 (as per the on-screen caption), more than 1½ years before Truman ascended to office. Vice President Henry A. Wallace was very much aware of the project to build an atomic bomb, and had chaired FDR's secret board in 1942 to determine if such a weapon were possible. Vice President Wallace is often regarded as the father of the Manhattan Project.
Sid Liao mentions "Interpol". While the organization was founded in 1923, it did not go by that name in 1943. It was originally known as the International Criminal Police Commission. It adopted INTERPOL as a telegraphic address in 1946, three years after the episode takes place, and did not become officially known as Interpol until the mid 1950s.
Dr. Winter addresses one of the MPs as "Sergeant First Class". That rank did not exist in the United States military prior to 1951.
Col. Alden Cox military dress uniform has ribbons that were not available in the time period. He is wearing the World War 2 Victory Medal, World War 2 Army Occupation medal which were not available until the wars end. He is also wearing the National Defense medal, the Korean Service medal and the United Nations Korean Service medal all which were not available until the 1950's.
Frank Winter has a nightmare where his wife and another woman walk into the flash and mushroom cloud of an exploding atomic bomb. While very dramatic, it could not happen since no one would know what an atomic explosion would look like before the first test explosion.