Who first understood the importance of poles?
If you consider poles of an integral transform domain to be important to the solution of differential equations: (as usual,) Euler did it first, 1753.
One "importance" of poles is that they're part of a very useful representation for linear systems. They must've appeared when people started looking at functions as built from generating functions, so that'd date that .
So, both Bode and Nyquist are probably 100 years too late: this would emerge from looking at systems like Laplace did, around 1785.
If you think of poles as something that is relevant to your analysis of holomorphic functions, then clearly Cauchy would be your guy, starting with "Sur les intégrales définies", 1814, where he looked at simple poles within an area of integration.
When you really look into where the residue theorem (which really concerns itself with integrals over functions with poles, and is immensely important to a lot of things that we can do with the Fourier transform) was formed, you'd be looking at Lindelöf's "Le calcul des résidus et ses applications à la théorie des fonctions", 1905.
Finally, the Nyquist stability criterion was published when he was working at Siemens, 1930.


     Laplace     Lindelöf
  Euler | Cauchy    |
    |   |  |        |  Nyquist
    v   v  v        v  v
|---------.---------.---------.--&gt;
1         1         1         2
7         8         9         0
0         0         0         0
0         0         0         0