I didn't say they have
atomic clocks. They do have clocks, though. Like most computers do. And they are high-precision, and are low-drift
enough to predict the locations of satellites as long as they have been re-synchronized within the last few days or so.
Which, as you say, also happens by just observing the time signatures from the satellites. You need four visible satellites to determine your own time, though, whereas you only need three for position, so time isn't re-synched as often as position is calculated. The internal clock in the receiver allows the receiver to carry on tracking with only three time sources for a while.
But, to be clear on the topic of the parent discussion: I believe JDAM missiles (the ones that actually do use GPS) do have either an atomic clock source [more recently], or [formerly] have at least a high-precision monotonic clock source with low drift that is synchronized at point-of-launch by the clock on the bomber, which also has an HPC that was calibrated at its launch by a real atomic clock. They don't need to rely on external time-sync.
And modern ICBMs? Well, unless your jammer/spoofer can keep up with them, or is itself a satellite, you're only going to be able to affect them when they're on their descent course and making final adjustments. And, like this article says (https://www.technologyreview.com/s/423363/how-cruise-missile...), ICBMs have redundant aiming systems based on computer vision applied to either visual-spectrum or radar-based sensors.