We explore the nonlinear optical response of 3D gyroidal metamaterials, which show >10-fold enhancements compared to all other metallic nanomaterials as well as bulk gold. A simple analytical model for this metamaterial response shows how the reflectivity spectrum scales with the metal fill fraction and the refractive index of the material that the metallic nanostructure is embedded in. The ultrafast response arising from the interconnected 3D nanostructure can be separated into electronic and lattice contributions with strong spectral dependences on the dielectric filling of the gyroids, which invert the sign of the nonlinear transient reflectivity changes. These metamaterials thus provide a wide variety of tuneable nonlinear optical properties, which can be utilised for frequency mixing, optical switching, phase modulators, novel emitters, and enhanced sensing.