What's the slowest nuclear decay rate that's been measurably linked with a narrow energy linewidth? As was explained in the question What is the relation between the half-time and the line-width of a radioactive nucleus?, the half-life $\tau$ of an unstable nucleus is related to the linewidth $\sigma_E$ of the resonance in its energy spectrum by the uncertainty relation $$\sigma_E\cdot\tau=\frac\hbar 2.$$ This is required by the Fourier relation between the two, and it has apparently been confirmed experimentally for relatively short decays.
How short are such short decays? What's the longest radioactive decay half-life that's been experimentally shown to coincide with an experimentally-observed linewidth?
 A: About short-decays it is not easy to measure resonance widths. There are many reasons for this. Short-lived resonances are wide and their ranges of energies overlap at least partially with other resonances. I mean, neighbor resonances form a strong background.
Therefore, comparisons with theoretical predictions are very difficult.
The ideal case for defining the resonance width is when the resonance is sharp, i.e. the width is much smaller than the central energy of the resonance, and also much smaller than the distance between the central energy in the resonance and the potential barrier. These conditions can be fulfilled by long-lived nuclides, however in this case the number of events detected in experiments is not big and the statistics is poor.
Thus, for your last question, the hope to give an answer is very small. The imprecision of the results of measurements is so big, that the half-life is usually given as log$(\text t_{1/2})$. And from one series of experiments to another series carried some years later (eventually by another team of experimenters), there may be differences of 10 times (e.g. see the report on the half-life measurement of $^{130} \rm Te$ ).
Moreover, in comparisons of the measured half-life with theoretical models, it is not the agreement with the resonance width that is tested, but the agreement with all sort of rules about the half-life, predicted by models, see for instance the following reference
G. Royer, "ALPHA DECAY POTENTIAL BARRIERS AND HALF-LIVES AND ANALYTICAL FORMULA PREDICTIONS FOR SUPERHEAVY NUCLEI", "Workshop on the State of the Art in Nuclear Cluster Physics (SOTANCP2008), Strasbourg : France (2008)".)
By the way, in cases that the law of decay obeys the exponential form $N(t) = N_0 \ e^{-\Gamma t}$
the relation between the half-life and the resonance width (usually denoted by $\Gamma$) is
$$ \Gamma \ \text t_{1/2} = \hbar \ ln\ 2$$
A: Just to offer a random benchmark--because I haven't made a study of this question--$^5\mathrm{He}$ is described as having a linewidth of $0.60 \,\mathrm{MeV}$, which corresponds to a lifetime in the single digits of nanoseconds.
References:

*

*http://ie.lbl.gov/toi/nuclide.asp?iZA=20005 The data for which is taken from the ENDFs.

*https://doi.org/10.1016/S0375-9474(02)00597-3 Nuclear Physics A 708 (2002) 3–163
"Energy levels of light nuclei A = 5, 6, 7" A review article. More than two dozen papers are cited as discussion $^5\mathrm{He}$ measurements. In a couple of cases the width is explicitly described has having been measured.

