Differently sized particles in sheared granular mixtures undergo size segregation and diffusive remixing, the relative magnitude of which controls the degree to which distinct particle layers form. The presence of viscous interstitial fluids affects individual particle dynamics, resulting in complex segregation and diffusion behaviors. In this study, the effects of different types of fluids (characterized by fluid density and viscosity) on these two processes are investigated, for various flow conditions and material parameters, via coupled numerical simulations of immersed granular shear flows. We observe that both the segregation velocity and the diffusion strength decrease with the fluid viscosity, but these effects occur only when the viscosity exceeds certain threshold values, indicating a transition from viscous to inertial regimes. In the low-viscosity limit where fluid and grain inertia dominate, both segregation and diffusion processes depend on flow conditions and material properties in a manner that is similar to those in dry inertial granular flows. On the other hand, decreasing the relative density between the particles and the fluid slows down segregation but does not significantly affect diffusion. Based on scaling analysis of the simulation data, empirical relationships for the segregation velocity and diffusion coefficient are developed as functions of a modified Stokes number, and are then used to extend an existing segregation-diffusion continuum equation for granular mixtures immersed in different types of fluids.