Can I cure UV adhesives without a UV light source?

Yes. UV adhesives are optimized to cure under UV light. UV adhesives absorb UV light and pass the visible and IR portions of the spectrum. However, since sunlight and most room lights also emit in the UV, the adhesives will cure under these conditions. Please note that initial and full cure times will be considerably longer, because there will be less energy spread over a larger area then when compared to a UV curing light source.