Time-varying media have regained strong interest across physics and engineering because they enable wave phenomena inaccessible in static systems, including temporal pumping/steering, temporal topological boundary states, frequency conversion and parametric amplification. When temporal modulation is combined with spatial variation to form a traveling-wave spatiotemporal profile along a waveguide, time-reversal symmetry is broken, thereby inducing nonreciprocal wave transport. Spatiotemporally modulated elastic metamaterials offer a compelling platform for nonreciprocal control. However, most existing studies assume continuous, effectively time-infinite modulation, leaving scattering at time interfaces (i.e., sudden switching events) relatively underexplored. In this work, we investigate longitudinal elastic waves propagating through a finite spatiotemporally modulated temporal interlayer bounded by two time interfaces. Across these interfaces, the wavenumber is conserved, whereas the frequency content is abruptly redistributed. Guided by Floquet–Bloch dispersion relations, we develop a mode-coupling theory that enforces displacement and momentum continuity at the interfaces, enabling prediction of multi-order temporal transmission/reflection and modulation-driven energy pumping. Two distinct regimes are revealed. In the subsonic case, directional frequency bandgaps yield nonreciprocal wave conversion, where opposite incidences lead to asymmetric conversion between transmission and frequency-shifted reflection. In the supersonic case, directional wavenumber bandgaps with complex frequencies produce nonreciprocal parametric amplification in both transmitted and reflected fields, again accompanied by frequency conversion. Finite-difference time-domain simulations quantitatively corroborate the predicted scattering spectra and wavefield evolution. Overall, the proposed framework clarifies time-interface scattering in finite-duration spatiotemporal media and supports the design of temporal elastic metamaterials for one-way filtering, amplification, and frequency conversion.