This doesn't answer the original question: How could the experimenter ensure the electromagnetic pulse from the nuclear denotation wouldn't create false readings in the TDR measurements?
Nuclear reactions don't just make EMPs; it's not like nuclear plants are emitting continuous electromagnetic interference. Nuclear EMPs are created when radiation ionizes susceptible atoms, creating charged particles via Compton scattering. That's why high-altitude nukes create much more intense EMPs; the low-density atmosphere is much easier to ionize.
Gases at low pressure are DRAMATICALLY easier to ionize than anything else- think fluorescent tubes, etc. Dirt will not be highly ionized by an underground explosion, although I can't actually find anything about underground EMPs. It is safe to say that the EMP will be very greatly reduced, though.
The EMP is dangerous to electronics because it creates a voltage gradient over a large area. A 5 volt gradient across a few inches of ground plane will destroy most electronics permanently. A semiconductor is painstakingly embedded with atoms that create an intrinsic voltage difference across a tiny sliver of silicon; a mild voltage in the wrong direction will erase that gap. It will not destroy wires, since the pulse is extremely short.
Forget erroneous measurements. The EMP would kill the computer if it was just a plain wire. You're talking about picking up tens of thousands of volts across any kind of protection circuit. The only possible way this works is with some protection, ideally shielded wire like a coax cable. If that's the case it'll just cause a brief (<10 microsecond, mostly) flash of current to ground. Modern Ethernet is twisted pair, but the first ethernet was coax and the first R58 was sometimes shielded.
There are other types and components of nuclear EMPs but they mostly aren't important in this case. There are several highly questionable things about this story- the PDP-11 would need a pretty skookum power supply to not bottom out, the motivation is... strange, the ethernet wire would need to be incredibly long, etc. etc. EMP is probably not one of them, at least as far as the quality of data. EMP becomes irrelevant if you can access raw data or the ping uses anything but the simplest impulse waveform. Assuming you can actually collect data in the first place, anyway.
It might -- in this context (and always ;) ), I'd guess that experiment is the arbiter of truth. If they set up a few of them, perhaps with differing cable lengths in order to vary the capacitance, the extent to which they all measured similar results would give a measure of how well the method might measure the cable ablation.
There's a chance that there's a subtle systematic, but my guess is that either the method would work or the measuring device would get smoked, with not much in between.
