IIRC, you have to have samples to match the particles with the source.
From what I remember, it's more about where the Uranium came from in the first place. Reactor samples would be the easiest way, but traditional detective work from the orgin of the Uranium would fill in the missing pieces.
Tracing bomb material to its source may be only the beginning of an investigation, rather than the end, as the authors acknowledge. Discovering that a terrorist explosive was made of uranium stolen from a specific site in Russia, for example, does not identify the terrorists, but it does provide a starting point, especially if there is suspicion that the bomb makers had inside help.
If you have the origin, you have a lead. And I'm sure after a post-Nuclear event the US or Britain or Russia would throw everything at establishing the trail to weaponisation. I'm also 100% sure that every help would be given to them to avoid incrimination. If, let's say, the trail lead to Syria or Iran, I think they wouldn't like to hinder the investigation. Anything other than complete and transparent cooperation would be causus belli in the eyes of the world.
The article makes the case for more resources for nuclear forensics. I'm surprised that this is an issue in the current climate.