I know this event, nearly 155 years ago, has been discussed before. But IMO the debate moved too quickly towards the future - 1870 and later.
Modified from the Wiki:
Paris, France: On the evening of 14 January 1858, as Emperor Napoleon III. and Empress Eugénie were on their way to the theatre in the Rue Pelletier, the Italian radical Felice Orsini and his accomplices threw three bombs at the imperial carriage. The first bomb landed among the horsemen in front of the carriage. The second bomb wounded the animals and smashed the carriage glass. The third bomb landed in the carriage and killed the emperor and empress.
Additional notes:
~ Orsini himself was wounded on the right temple and stunned. He tended to his wounds and returned to his lodgings, where police found him the next day.
~ French police quickly determined that the bombs had been manufactured in England.
~ The heir, Napoleon, Prince Imperial, is not quite two years old.
~ The most influential Bonaparte will be Prince Napoleon, known as Plon-Plon. He is an anti-clerical liberal and favors the unification of Italy and the abolition of the Papal State. (Since this is before the 1859 Italian War, Plon-Plon has not yet the reputation as a battle-shy coward.)
What I don't know: How popular will be Bonapartism in general in early 1858? The Emperor has been martyred, but his heir is extremely weak. I cannot imagine much sympathy for the Italian cause following the assassination, and that might severly hamper any attempt of Plon-Plon to gain power.
How likely is another revolution? The idea of a republic will be very alive.
Other claimants are:
~ The Legitimist pretender of the time is Henri, the Count of Chambord. He lives in Schloss Frohsdorf in Austria.
~ The Orleanist pretender of the time is Philippe, the Count of Paris. He lives in Sheen House in Surrey, England.
~ French monarchists try to create a compromise by having the childless Chambord as first pretender and the Count of Paris as his desiganted heir, but nothing is resolved.
General ideas (German-centric, I admit):
*If* there is no French-Sardinian alliance, there will be no Italian War. This will highly affect the International Red Cross.
If there is no Italian War, there will be no fear of a French move against Austria proper through Southern Germany, thus no need for a general mobilisation in the German states.
Without this mobilisation, Prussia does no realite how weak their current system is and how urgently they need a military reform.
Without the royal reform plans, the conflict with the liberal parliament will not happen (or not anywhere as bitter), so the last-effort appointment of Bismarck will not happen.
There are many parts of the globe that will be similarly affected - Indochina, Algeria, Mexico, but I do not know enough about them to speculate.
But the central question is, of course: What will probably happen in France itself in 1858 and 1859?