We can model the flame of a wood fire as a black body radiator, since the glow of the flame largely comes from the heated solids given off from combustion.
A blackbody radiator emits light purely as a function of its temperature: as the object gets hotter, it emits shorter wavelengths of light. The sun is somewhere around 5500 Kelvin, which peaks in the visible spectrum, but gives off a significant amount of UV radiation, as well as a massive amount of IR radiation and lower.
A wood fire only burns at about 1500K, which has a peak in the IR spectrum, and pokes into the Visible spectrum, which is why wood flames appear orange/red. The blackbody curve is virtually 0 at the green wavelength, and is even smaller in the UV. That means that almost no UV radiation is given off by a wood fire.
You can play around with this Blackbody Curve model by adjusting the temperature, and seeing what wavelengths an object gives off at different temperatures. https://phet.colorado.edu/sims/blackbody-spectrum/blackbody-spectrum_en.html
Sidenote: this is why night vision cameras use IR to see in the dark. At room temperature, blackbodies give off a lot of radiation in the IR frequency, so IR cameras can pick up those wavelengths even though nothing is emitted in the visible spectrum.