My laser pointer has a wavelength of 405nm so it's very close to UV.u
How I think it works is that the photons incident on the phosphorescent material excite orbital electrons to a higher energy level, then at a later time the electrons fall back to their original energy level emitting a photon of visible light.
phosphorescent pigment for analogue watches, fire escape signs etc. is optimised to have the emission of the photons delayed by quite a considerable time so it'll continue to glow for a few minutes after being exposed to light - tbh not sure on the detail of how the delay is achieved.
Having mucked around with some fire escape signs at work I think there is a threshold wavelength for incident light to excite the electrons (similar principle to the photoelectric effect) in that you can blast the material with a red laser and get no light emitted after you stop... but it'll emit light strongly after you hit it with a blue LED (built into a poundland cigarette lighter) - so it's evidence for the particle nature of light.
an entirely legitimate use of phosphorescent pigment is making fake radioactive waste spills... like this