So far as the position of today's British royal family is concerned, no person born illegitimate has any right of succession to the British throne. If an illegitimate claimant appeared, then, according to my understanding, all he or she would be entitled to would be to petition for financial support if still a minor.
I think that the days of being given a surname of Fitzsomething and a Dukedom are long-gone!!
I am not expert in European Royal and Princely families, but from memory, Prince Albert of Monaco has faced several paternity claims over the years. I believe that in at least one case, Prince Albert has admitted paternity and paid child support. I think that the settlement provided that there was to be no right of sucession to the Princely throne. Someone more expert than me on Monaco's reigning family would be better placed to assist, I feel.
Various claims of illegitimacy are not unknown in the recent past of the British Royal Family, although succession has never been the issue. The Duchess of Cornwall's great grandmother, Alice Keppel, was the mistress of Edward VII and may or may not have borne him any children. The Television Annoucer, Mary Malcolm, who died quite recently was the grand daughter of Lily Langtry, mistress of Edward VII, the implication being that Mary Malcolm was the illegitimate grandaughter of the King. There has never been any question of any illegitimate offspring the King might have had claiming succession to the throne.
The situation with regard to Prince Albert's father Prince Rainier is also interesting. Prince Rainier's mother, Hereditary Princess Charlotte, Duchess of Valentinois was born illegitamely. She was the only child of Prince Louis II and she was later legitimised through formal adoption and subsequently named heiress to the throne of Monaco. The Monagasque succession therefore extends to legitimsing offspring, which is not permissable under British Laws.
I am sure that someone better qualified than me can help you.
Good Luck
Alex.