Without Henri, Comte de Chambord, the mainline of the Bourbons will die with Louis, Comte d'Angoulême, the eldest son of Charles X, as he was infertile. Then comes the question of who's the third-in-line and there are the two possibilities linked to the question of Utrecht:
- On the Legitimist side (Salic Law applied without Utrecht), the heir would be King Ferdinand VII of Spain and after him his brother Infante Don Carlos.
- On the Orléanist Side (Salic Law applied with Utrecht), the heir would be Louis-Philippe, Duke of Orléans.
Regarding the political effects of the absence of "Henri V", I suspect there might be not much important. The 1830 Revolution wanted Charles X out and "Louis XIX" abdicated 20mn after his father. Maybe Louis, Comte d'Angoulême, could be tempted to become King of France but I doubt it would work. In fact, I don't see how this plays on French politics until 1844 (death of Louis, Comte d'Angoulême) as the main bloodline of the Bourbons would be extinct: this would probably cause an earlier ralliement of the majority of the Legitimists to the Orléanists. But even then, it is still a long way to go: 1848 Revolution is probably not butterflied away, as would be the Second French Empire... This might increase the chances of a Monarchist Restoration after the Franco-Prussian War though.