Everyone is familiar with "Wind Chill Factor," which is an estimate of the effect of air movement on the speed with which ambient cold air will chill exposed human skin. Wet bulb temperature readings are likewise dependent on air movement to equalize the RH surrounding the wick with the RH of the chamber as a whole--within a reasonable length of time. The upper limit of effect (lowering of the wet bulb temp due to air movement) is merely that the difference between the wet bulb temp and the dry bulb is a true indicator of ambient humidity at any given time. Extra air movement (containing the same RH) won't reduce the wet buld temp any lower.
With zero air movement, the air immediately surrounding the wick covering the wet bulb becomes saturated, reducing the evaporative cooling of that thermometer. Even with hurricane force wind, the air immediately surrounding the wet bulb wick can only evaporate water into the same ambient air--it just gets there faster.
I suspect that simple air convection in a heated chamber is sufficient to give a good wet bulb reading +- a few minutes.
Bob