I was doing some research on the Great Seal of the Realm, and noticed that until the Acts of Union under George III, the English or British monarchs had been styled 'of France'. George III was, for example, George III, by the grace of God, of Great Britain, France, and Ireland, King, Defender of the Faith.
After the Acts of Union, he adopted the style we commonly associate with popular monarchies: George the Third, by the grace of God, King of the Britons, Defender of the Faith.
So, why was this claim dropped? I thought any practical claim would have ended after the loss of Calais (even that being tenuous). I know the Jacobites sheltered in France, and I can't imagine the French king being too happy about putting up a pretender to his throne.
Well, the French throne ceased to exist in 1792, when the French Revolution overthrew the monarchy. During the negotiations of the Treaty of Lille, in 1797, the French demanded that British monarchs drop their claims to the throne of France. When the Act of Union of 1800 joined the Kingdom of Ireland to Great Britain, George III used the opportunity to drop the claim to the throne as it was now defunct, and Britain formally recognized republican France in the treaty of Amiens in 1802. As to the Jacobites, the claimant at the time was Cardinal Henry Benedict Stuart, who did not drop his claim until his death in 1807. Being a Roman Catholic bishop, he left no offspring and was the last of the Stuart claimants.