The axion insulator exhibits a topological magnetoelectric effect characterized by an axion angle $\theta =\pi$, while the time-reversal-doubled axion insulator (T-DAXI) can be viewed as two copies of an axion insulator related by time-reversal symmetry. In this work, we show that a nonsymmorphic topological crystalline insulator with glide (or screw) symmetry can host the nonsymmorphic T-DAXI phase, diagnosed by the nontrivial hourglass (screw) invariant ${\delta }_h=1$ (${\delta }_s=1$). In this phase, the partial axion angles (${\theta }^{\pm }=\pi$) are quantized by the glide or screw symmetry, and the gapped surfaces realize half-quantized quantum spin Hall states. Furthermore, we show that nonsymmorphic T-DAXIs also exhibit the spin-magnetoelectric effect. Namely, an external magnetic field can induce a topological nonlocal spin polarization in real space. When the magnetic field is time-periodic, this leads to an alternating spin current detectable in experiment. Using ab initio calculations, we demonstrate that mixed bismuth monohalides Bi4BrxI4−x realize the nonsymmorphic T-DAXI. Our findings not only reveal the symmetry-enforced T-DAXIs in nonsymmorphic topological crystalline insulators, but also demonstrate the spin-magnetoelectric effect as a detectable topological spin response in nonsymmorphic T-DAXIs.