Carbonaceous slate was taken as the research object and creep mechanical tests were conducted on carbonaceous slate with different bedding dip angles in response to the anisotropic characteristics of layered rock masses. An improved Nishihara nonlinear damage creep constitutive model that can describe the accelerated creep of carbonaceous slate with different bedding angles was established based on the creep test results. The one-dimensional and three-dimensional constitutive equations of the model were derived. Results show that there is a significant stress threshold in the creep process of carbonaceous slate. Only attenuation creep occurs in carbonaceous slate when the stress level is less than the threshold. Steady-state creep begins to occur when the stress level reaches or exceeds the threshold. The carbonaceous slate undergoes accelerated creep and creep failure when the stress level reaches or exceeds the failure stress. The inversion identification results of model parameters indicate that the improved Nishihara creep model can effectively describe the entire creep process of carbonaceous slate. The relationship between model parameters and confining pressure and bedding angle was discussed based on the parameter identification results. The relationship between damage parameter c and layer inclination angle was obtained. The effects of damage parameter d and viscosity coefficient to accelerate creep stage were analyzed.