We focused on developing polyurethane (PU) adhesives with superior ambient thermal and hydrolytic stability, a crucial factor for industrial productivity. Our approach involved creating PU prepolymers that can withstand varying temperatures in ambient conditions. These prepolymers consist of conventional isocyanate-terminated polyurethane and metal-free acid:base organic catalysts, with the stability of the adhesive relying on the organocatalyst employed. We tested a series of 11 latent organocatalysts derived from the reaction between 1,8-diazabicyclo[5.4.0]undec-7-ene (DBU) and various acids. Among these, the catalyst based on 1-naphthoic acid exhibited exceptional stability, lasting at least 3 h at 60 °C and an average relative humidity of 65% under vigorous stirring. We assessed this stability using a fan-based stirrer and analyzed the curing conditions kinetically through DSC. Furthermore, our adhesive formulation is environmentally friendly as it is free of metals, specifically tin (typically present in catalysts such as dibutyltin dilaurate). This quality enhances its sustainability. To validate the practical applicability of the adhesives, we conducted tests using decorative facade models composed of siliciclastic sandstone extracted from a quarry in Vilviestre del Pinar (Burgos, Spain. Latitude: 41.951024° N, longitude: 3.078283° W) and extruded polystyrene (XPS). The results demonstrated the excellent hydrolytic and thermal stability of the adhesives, highlighting their significant potential for panel manufacturing in this context.