Nodal-line semimetals are topologically nontrivial states of matter featuring band crossings along a closed curve, i.e., nodal-line, in momentum space. Through a detailed analysis of the electronic structure, we show, for the first time, that the normal state of the superconductor NaAlSi, with a critical temperature of T_c ≈ 7 K, is a nodal-line semimetal, where the complex nodal-line structure is protected by nonsymmorphic mirror crystal symmetries. We further report on muon spin rotation experiments revealing that the superconductivity in NaAlSi is truly of bulk nature, featuring a fully gapped Fermi-surface. The temperature-dependent magnetic penetration depth can be well described by a two-gap model consisting of two s-wave symmetric gaps with Δ₁ = 0.6(2) meV and Δ₂ = 1.39(1) meV. The zero-field muon experiment indicates that time-reversal symmetry is preserved in the superconducting state. Our observations suggest that, notwithstanding its topologically nontrivial band structure, NaAlSi may be suitably interpreted as a conventional London superconductor, while more exotic superconducting gap symmetries cannot be excluded. The intertwining of topological electronic states and superconductivity renders NaAlSi a prototypical platform to search for unprecedented topological quantum phases.