While the main emissions from these fires are CO2 and CO, the more important difference is what happens to the poorly combusted carbon products that are aggregated into larger particles. The climate community tends to refer to these as black carbon (BC), although that’s an oversimplification. These BC particles are strong absorbers of solar radiation, causing local atmospheric heating, but they also have a very short lifetime in the troposphere (~5 days; Baker et al, 2015), so the amount of BC in the atmosphere at any one time is relatively small (~0.1 Tg). This has an overall warming effect on the troposphere, which is why it’s included in calculations of historical and future greenhouse warming, but it’s not as large as that of a long lived and abundant species like CO2.
An important thing to note is that there are large scale differences between current, normal smoke emissions and those from a nuclear incident. Current BC emissions from all sorts of surface burning processes are about 7.2 Tg / yr (~0.02 Tg / day; Kilmont et al, 2017). A big number compared with that ~0.1 Tg atmospheric burden but, as I say, the removal processes are fast. These emissions are spread over most of the global land area (although they’re greatest in Africa, India and China), so the emissions per unit area are also relatively low.
In a single nuclear incident the BC emissions would be much greater, would occur in a small area and would be injected throughout the troposphere. For example, Robock et al (2007), the same people you link to in your question, ran this simulation:
In our standard calculation, we inject 5 Tg of black carbon on 15 May into one column of grid boxes at 30 N, 70 E. We place the black carbon in the model layers that correspond to the upper troposphere (300–150 mb).
This is a huge perturbation - about 50 times the current atmospheric total and about 250 times the current daily flux. They found that the BC entered the stratosphere, where removal processes are much slower than in the troposphere, such that,
E-folding times are 6 y, compared with 1 y for volcanic eruptions and 1 week for tropospheric aerosols.
The BC then absorbs solar radiation and heats the stratosphere, reducing the amount of solar radiation reaching the surface and cooling surface air temperature by more than 1 degC for about 5 years. Note that the associated CO2 emissions would warm the troposphere, but over a slower time scale and these studies tend to concentrate on the immediate BC cooling effect of a nuclear incident.
As an additional bit of contrast, that study also found that,
When we placed the aerosols in the lower troposphere (907–765 mb), about half of the aerosols were removed within 15 days
This indicates how important it is for the BC emissions to reach the stratosphere in order get a prolonged surface cooling.