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In the early 60s, military computers were small enough to be built into missiles. Transistors were invented in the early 50s and just a few millimeters tall each in in the early 60s – you could concentrate thousands of them in a small, light package and build any kind of logic circuits with them. Magnetic-core memory was already able to store several Kilobytes in those days. Everything was expensive and hand-built, but that doesn’t really matter for the military at the height of the Cold War, does it?
For guidance, the most important component is the gyroscope, which controls the orientation of a rocket or plane and can be used to directly change the angle of ailerons and fins, like the V2 did it in WW2, or input those informations to those early computers for more nuanced control – in combination with accelerometers, of course. Early intercontinental missiles such as the first few Atlas models in the 50s relayed this information to ground stations via radio where the calculations would be made, but later on miniaturization had progressed far enough to allow for fully automatic rockets. In the end, you just need a simple program that controls the flight on a pre-calculated route, cuts the engine(s) at the right point in time and ignites the payload.
What’s important to understand is that early 60s ICBMs were not tremendously accurate – they didn’t have to. Their warheads were already 100x as powerful as the Hiroshima bomb. The blast radius effectively compensated for the missiles’ limited accuracy. This of course limited their use to attacking centers of population, while fleets of much more accurate bombers would destroy smaller military targets. Every nuclear power of the Cold War adhered to this strategy and even today ~~has~~ [Edit:] some of them have both missiles *and* bombers with nuclear warheads/bombs.