We consider nonparametric statistical inference on a periodic interaction potential $W$ from noisy discrete space-time measurements of solutions $ρ=ρ_W$ of the nonlinear McKean-Vlasov equation, describing the probability density of the mean field limit of an interacting particle system. We show how Gaussian process priors assigned to $W$ give rise to posterior mean estimators that exhibit fast convergence rates for the implied estimated densities $\bar ρ$ towards $ρ_W$. We further show that if the initial condition $φ$ is not too smooth and satisfies a standard deconvolvability condition, then one can consistently infer Sobolev-regular potentials $W$ at convergence rates $N^{-θ}$ for appropriate $θ>0$, where $N$ is the number of measurements. The exponent $θ$ can be taken to approach $1/2$ as the regularity of $W$ increases corresponding to `near-parametric' models.