Summary: The paper is concerned with the convergence analysis of a numerical method for nonlocal Cahn-Hilliard equations. The temporal discretization is based on the implicit midpoint rule and a Fourier spectral discretization is used with respect to the spatial variables. The sequence of numerical approximations in shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme are derived. The uniform bounds on the sequence of numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial discretization parameter needs to be assumed.