Recent experiments reported that proton mobility in tetramethylurea (TMU) solutions is much slower than in urea solutions of the same molarity, and this (as well as the significantly retarded water reorientation) was ascribed to hydrophopic effects. In order to further explore the mechanism of proton transport in these solutions, reactive molecular dynamics simulations using a multistate empirical valence bond model were conducted. The simulations showed that the hydrophobic effect of the TMU methyl groups is weaker than believed. Rather, water concentration is the dominant factor determining proton diffusion. This contrasts with water reorientation and selfdiffusion in these samples, which are mutually correlated and depend on the identity of the solute. Interestingly, we find that the mean squared displacements (MSDs) of both water and proton grow nonlinearly in time up to at least 1 ns ("transient subdiffusion"). Subdiffusion is more pronounced for the proton, with an exponent as low as 0.85 that depends, again, on water concentration. Hence, the experimentally relevant long-time diffusivity is observably smaller than what is usually deduced from short simulation runs. It exhibits, for both water and proton, a universal dependence on the power-law exponent.