Both superpowers basically used guided missiles with a manned capsule on top rather than a war head. How did they tell the difference between an ICBM vs. a manned space launch? I'd assume they would have some communication with the other side, but I'm sure they'd take it with a grain of salt... I mean "Yup, it's a space launch. hee hee" and have it be a nuclear missile headed for Moscow doesn't sound too far fetched after hearing some of the other things they considered.
Just curious after seeing a Gemini rocket taking off in /r/space.
A single missile would be a poor way to start World War III, any nuclear first strike would have been massive and targeted at destroying the other country's strike capabilities rather than cities/conventional forces. This would reduce the damage the other side could do in their response strike and allow the second/third waves to target and theoretically incapacitate enemy leadership and conventional forces. This is why the U.S. had the strategic triad of missiles, bombers, and subs. Even if the Soviets destroyed all the missile sites the U.S. could deliver an equally devastating strike from assets that the Soviets did not know the location of (sub underwater, bomber already airborne) reducing the appeal of a first strike.
A real life example of a scare about a launch comes from the NATO Able Archer exercise in 1983. The exercise occurred at a low point in U.S./Soviet relations and during it the Soviet early warning radar erroneously detected the launch of a single missile. The Soviet officer in charge of the system dismissed the positive signature as an error in the radar, as launching a first strike with only one missile was essentially committing national suicide for the attacker.
An incident in 1979 demonstrates what a likely attack would look like. Carter's National Security Advisor Zbigniew Brzezinski received an early morning call informing him that over 200 missiles had launched from the Soviet Union and were inbound towards the U.S. The next report indicated that over 2000 missiles were on the way. Brzezinski was about to call Carter to get him to order a counter-strike when he received word that it was an error. The glitch was blamed on a training tape being left in the system, but was actually the result of a software error. The National Security Archive has a good collection of papers on the incident available online.
TLDR: Space launches weren't a concern because a single missile was highly unlikely to represent the start of World War III
They communicated their mission launch schedules with each other. Plus, the American launches are never secret and are always made available to the public well beforehand.