How can sodium and potassium from burned organic matter (ashes) reenter the ecosystem?
Wood ash is commonly used as a fertilizer. It acts this way in natural burning ecosystems also, where the ash from burned plants fertilizes the growth of new plants and those that survive the wildfires.

The primary constituents of wood ash are calcium and potassium salts. Lye is extracted from these salts by extracting them in water; the hydroxide salts are more soluble so the other salts are left in lower concentration than the original ash.

However, when in solution, KOH isn't really a thing anymore: you just have potassium and hydroxide ions floating around separately. Nothing special needs to be done to separate the potassium from the hydroxide ion except dissolving in water. Neutralizing the solution is then just a matter of adding an acid. There are plenty of sources of acid in soil, and rain itself tends to bring acidity. One of the sources of acidity in rain is, in fact, from fires. It is not as strong as the 'acid rain' caused by industrial sources, but there is no magic in fire that causes basic chemicals to be left behind without producing acid ones elsewhere.

The main damaging effects of wood ash are if it is applied in high concentrations where it can directly enter water sources (see for example UGA extension). In this case, it can raise pH enough to be damaging to life. Large quantities from large fires can also raise pH substantially in streams and such (see for example Burton, C. A., Hoefen, T. M., Plumlee, G. S., Baumberger, K. L., Backlin, A. R., Gallegos, E., &amp; Fisher, R. N. (2016). Trace elements in stormflow, ash, and burned soil following the 2009 Station Fire in Southern California. PloS one, 11(5), e0153372.).

Eventually, however, there are enough sources of acid in the environment that there is no way the salts are permanently alkaline from the perspective of the whole environment.