Does the wire diameter of the reflector and the directors of a beam antenna affect the bandwidth or is it only the driven element diameter that has this effect?
This is a non-mathematical answer.
The only difference between the driven and non-driven elements is the connection of a feedline. Each of the elements resonance and position contribute to the characteristics of the antenna.
If the directors and reflectors are not of similar bandwidth to the driven element then they will not have a consistent phase and amplitude relationship with the driven element over the frequency range. The radiation pattern depends on the summation of the energy from each of the elements. The radiation pattern will change.