Yes, once you go below about 1A, its difficult to reliably mount and connect to a thin enough plain tinned copper wire, so low current fuses must use other wire compositions, and IMHO, its probable there are special purpose higher current fuses (other than thermal fuses) where the required I2t characteristics favor the use of alloy or non-copper wire, but I would expect that plain tinned copper wire is still dominant, if only because of its well understood fusing characteristics, acceptable mechanical strength at elevated temperatures, and that the relatively high melting point decreases the effect of ambient temperature on its fusing current.
Sorry I don't have sources handy - I last looked into this in detail many years ago, and even in this internet age, reputable fuse manufacturers aren't keen to publicly disclose their 'secret sauce'.