Basically, it shouldn't matter. If the local population treats this person like a royal, then you should a.) show respect accordingly, to get along better with the locals; and b.) perhaps even make an effort to see how royalty is treated. (The latter is an interesting halachic point for some other time.) Both of those points are about de facto, who is regarded as king. The person's technical halachic lineage really shouldn't matter.
Take the eldest son of the Last Shah of Iran. He's a Pahlavi, he lives in Virginia. There is no halacha that says "respect all Pahlavis." The halacha says "respect whoever the locals say should be respected!"
If we went with strict halachic definitions of lineage, for instance, then His Majesty Charles III is a Mountbatten (his father's side), not a Windsor (his mother's side), as non-Jews' lineage is strictly patrilineal. The royal family are the Windsors. Does that make him any less king? Absolutely not! If the UK says he's the king, then he's the king!
The one interesting case that would come up is what bracha you say upon seeing them. A non-Jewish head of state warrants the blessing "[G-d, who] gave of His honor to flesh and blood." A Jewish head of state gets "[G-d, who] shared some of His honor with those who revere Him." The halacha here does note that the latter phrasing applies to any observant, G-d fearing Jew who is a head of state. It doesn't matter that their state is nowhere near Israel, nor that their subjects are mostly non-Jewish. (E.g. in some alternate reality where Joe Lieberman became the US President.) But on this last point, the blessing is reserved for the head of state, so a prince or the like wouldn't trigger it regardless.