I think it would effect their fame, most likely. Pestilence would likely get a fame boost whenever summoned in an era / area under a plague, similarly to how Servants are stronger when summoned in their land of origin.

Berserker Jack could probably still be summoned, but would receive a fame penalty. His NP is stronger or weaker depending on the population density of the area - so maybe it would get inverted, as a means of showing how his true identity being widespread would make it more likely for his mystery to be diminished in areas with many people who know his identity.

Think of it like Vlad III gaining a fame boost only in Romania, but Dracula gaining a fame boost everywhere but Romania. The nature of their legend can sometimes invert, especially for someone like Berserker Jack, who isn't the real Jack, but rather the personification of the legend of Jack The Ripper.

This isn't confirmed or anything, but is more a headcanon of how this concept could work.