This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $Ψ(-Δ)$, where $Ψ$ is a Bernstein function. As applications, the existence and uniqueness of solutions to the corresponding nonlinear martingale problems are proved. Furthermore, it is shown that these solutions form a nonlinear Markov process in the sense of McKean such that their one-dimensional time marginal law densities are the solutions to the nonlocal nonlinear Fokker-Planck equation. Hence, McKean's program envisioned in his PNAS paper from 1966 is realized for these nonlocal PDEs.