I would expect that some of the predicted effects of a "nuclear winter" would start to appear after a series of forest fires--albeit smaller in scope and intensity--but AFAIK this is not the case. What gives?

They aren't usually big enough to get the particulates high enough. But big events can do it:

https://en.wikipedia.org/wiki/Year_Without_a_Summer

Most forest fires don't create strong enough firestorms (if they do at all) to funnel material high enough to have an effect -- most of the ash falls back down in a matter of weeks (even sooner if it gets rained out of the atmosphere). I'd compare nuclear blast firestorms with volcanic eruptions -- we do have a record of eruptions funneling material high enough into the troposphere and stratosphere to have a noticeable, longer-term effect on the climate (over the subsequent months and years).

There have been "volcanic winter" events. Or at least, reduced insolation for a year or more.