One of my ancestors was a member of the Royal Family of England

An illegitimate member, of course.

This was hardly surprising since illegitimate children (of all kinds) were common for most of history. If the mother and father had been living together for some time exclusively, parentage was assumed. 

This was the case with Julius Ceasar and Cleopatra, whose children were well known. 

Who was this person in my case? 

Some distant relatives of mine (the Fishers) have researched this and come up with a probable answer. But my closest relatives have kept this information for themselves.