Would optical signal processing still suffer from slew rate?
Your question includes both the conversion (since you speak of processing) and light propagation. Conversion involves electronics, as @Nasha mentions, and thus is directly impacted by the slew rate. 

Light propagation speed is reduced (with respect to that in vacuum) by the refractive index of the material. The physics causing the finite slew rate is also also causing the fact that the refractive index of the material is always greater than 1.