Radionuclides give off unique spectral signatures. I-131 looks different from Tc-99m (another common medical isotope) which looks different from cobalt-60 (an industrial isotope) which looks different from uranium.

Spectral signatures are the most obvious way which one would hope to differentiate nuclides. This is relatively easy in the laboratory environment. However, with all of the material between the source (imagine it inside a cargo container) and the detector (somewhere outside), the original emitted spectrum is significantly washed out. In other words, the sharp peaks are no longer sharp, and may not even be distinguishable peaks at all.

Even if we could distinguish uranium and its decay chain daughters, those nuclides are common sources of background radiation. Thus, we still have problems with false positives.

The timely detection of special nuclear materials is a challenging problem. The US is pouring a lot of money into research towards this end, but we are far from having a workable solution. The detector technology must continue to improve (CZT is a step in this direction). But in the end, algorithms and methods will likely play an indispensable role. So while the suggestion of putting a detector in every phone is entirely impractical with today's technology, it may hint at a potential solution to the problem.