Article briefly talked about delivery, which is tricky to do precisely at best of times, but didn't really mention how to address delivery into a nuclear blast. Hundreds of meters behind the craft about once a second doesn't seem like it would be enough time for the blast to clear so would get in the way of sending a new capsule backward. Anyway I'm sure it's just an implementation detail
A lot of the work was done to a design point of 0.25 seconds, and Dyson's book says the issue there wasn't the blast clearing, but just being able to move the machinery fast enough. I kind of share your puzzlement; I can see the blast clearing this quickly in space (the debris moves at tens of thousands of km/sec) but not in the atmosphere.
The blast clears that quickly in the atmosphere because the shockwave and debris move at-or-faster-than the speed of sound, so a few hundred meters is tenths of a second or less. You sit inside the mushroom cloud, of course, but the important part is gone quickly.
The Rapatronic camera was used to take these kinds of pictures, and you can see that the actual blast front is around 20 meters across after 1 millisecond (!!) https://en.wikipedia.org/wiki/Rapatronic_camera
In a vacuum, there isn’t a fireball hanging around for the next charge to cross; the plasma is moving outward at thousands of km/s.