We combined a spin-resolved photoemission spectrometer with a high-harmonic generation (HHG) laser source in order to perform spin-, time- and angle-resolved photoemission spectroscopy (STARPES) experiments on the transition metal dichalcogenide bulk WTe$_2$, a possible Weyl type-II semimetal. Measurements at different femtosecond pump-probe delays and comparison with spin-resolved one-step photoemission calculations provide insight into the spin polarization of electrons above the Fermi level in the region where Weyl points of WTe$_2$ are expected. We observe a spin accumulation above the Weyl points region, that is consistent with a spin-selective bottleneck effect due to the presence of spin polarized cone-like electronic structure. Our results support the feasibility of STARPES with HHG, which despite being experimentally challenging provides a unique way to study spin dynamics in photoemission.