Most tests have been performed underground. And for an above-ground test to cause anything close to a nuclear winter, you need a lot more material than the bomb itself and the soil beneath it. Tons and tons of debris would need to be shot up into the atmosphere.

So, the theory of nuclear winter depends on bombs detonating over cities and forests, causing massive amounts of debris, and also inducing firestorms that would act as chimneys for even more smoke and particles to be funneled up into the troposphere and stratosphere.

Because those were most all carefully controlled to prevent such damage. Top of the list: those that were above ground were tested above empty terrain. The bomb doesn't create most of the ash. The burning city under the bomb is where the bulk of sun-blocking ash comes from. No city, no ash, no nuclear winter issues.

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.