The recent discovery of high-T_c superconductivity in bilayer nickelates has highlighted the crucial role of interlayer coupling and motivated proposals for dominant interlayer pairing. Whether such pairing can support topological superconductivity, however, remains unexplored. Here, we demonstrate that substrate-supported bilayer nickelate thin films can host an intrinsic time-reversal-invariant topological superconducting phase protected by diagonal mirror symmetry. Using a two-orbital model derived from first-principles calculations, we show that a modest layer-asymmetric potential and layer-staggered Rashba spin-orbit coupling drive the system into this topological phase over a broad parameter range with a dominant interlayer pairing. The nontrivial topology arises from a reversal of the pairing sign on one of the spin-split bonding Fermi surfaces and is characterized by a nonzero mirror winding number along the diagonal direction, as well as a Majorana Kramers pair at the [11] edge. The phase also features bulk gap nodes located slightly away from the diagonal directions, featuring chiral charge and zero-energy flat bands. We discuss experimental routes for realizing and tuning this phase. Our results establish high-T_c bilayer nickelates as a promising platform for topological superconductivity, where tunable topological transitions and Majorana boundary modes can provide probes of unconventional pairing.