This paper provides both numerical and closed-form uniform asymptotic representations of the aperture Green's function which determines the magnetic field due to a point magnetic current source where both the source and field points lie on the ground plane which is covered by a planar material slab. The closed-form asymptotic representation which explicitly contains effects of space, surface, and leaky waves is shown to agree with the numerical evaluation of the exact integral representation for source and field point separations which are only a few tenths of a free space wavelength. An extended envelope extraction technique allows an efficient evaluation of the exact integral representation for source and observation separations too small for the asymptotic solution to be valid. The asymptotic solution is uniform in that it remains valid across all of the relevant surface and leaky wave transition regions. The computation time for this asymptotic result is 2--3 orders of magnitude faster than the numerical evaluation of the exact integral form, and additionally its computation time is independent of the source and field point separations. This aperture Green's function is very useful when computing the mutual coupling elements of the method of moments admittance matrix for an array of slots in a material covered ground plane.