  We measured the temporal voltage response of NbTi superconducting filaments
with varied nanoscale thicknesses to step current pulses that induce
non-equilibrium superconducting states governed by a hot-spot mechanism. Such
detected voltage emerges after a delay time td, which is intimately connected
to the gap relaxation and heat escape times. By employing time-dependent
Ginzburg-Landau theory to link the delay time to the applied current, we
determined that the gap relaxation time depends linearly on film thickness,
aligning with the acoustic mismatch theory for phonon transmission at the
superconductor-substrate interface. We thereby find a gap relaxation time of
104 ps per nm of thickness for NbTi films on polished sapphire. We further show
that interfacial interaction with the substrate significantly impacts the gap
relaxation time, with observed values of 9 ns on SiOx, 6.8 ns on fused silica,
and 5.2 ns on sapphire for a 50 nm thick NbTi strip at T = 5.75 K. These
insights are valuable for optimizing superconducting sensing technologies,
particularly the single-photon detectors that operate in the transient regime
of nanothin superconducting bridges and filaments
