Well, you'd need on that one to get back to 1879 at least I think.
To begin with, though there was a brief Bonapartist upsurge in 1919, with the election of about two dozen Bonapartist leaning deputies in the right wing landslide during the legislative elections, the movement largely floundered thereafter. Plus, after the death of Napoléon Victor, his son Napoléon Louis wasn't so keen on supporting a movement that was by this point hard on the right and it largely vanished after the 1930s and the government crackdown on far-right after 1934. Of note, the head of the Bonaparte House, Louis Napoléon later engaged in the Foreign Legion and fought in the Resistance against German occupation.
But besides that, the Bonapartist movement was already on severe decline. After the death of Napoléon III in 1873, the Bonapartist mantle was subject to a power struggle between the two competing main wings, which had already its roots back to the Second Empire days. The right wing of the Bonapartists, conservatives and clericals, was organized around Dowager Empress Eugénie and the Imperial Prince, aka Napoléon IV. The left wing meanwhile, the Jeromists, were organized around Prince Napoléon, Napoléon III's despised cousin, a noted anticlerical, supported by a number of liberals and even socialists, and closely aligned with Republicans' positions. This may be a simplification since both wings grouped factions that were susceptible to go back and forth between these two wings.
At this point, the Prince Impérial was still the undisputed heir and maintained the movement united around his persona. The Bonapartists then had a relatively good time. Despite the fall the Empire, they maintained a strong rural presence and while the Royalists were in full disarray amidst the elections of 1876 and 1877, after which MacMahon was forced to resign, the Bonapartist parliamentary group, the "Appel au Peuple" (Appeal to the People) broke through, becoming the main opposition to the Republican majority: while the Monarchists went down from 396 seats in 1871 to just 55 in 1877, the Bonapartists rose from 20 to 104 in the same time. The Prince Impérial had found considerable appeal in France and generated enthusiasm; he followed his father's footstep, taking up his populist streak, with noted pro labor positions (remember the right to strike in France was legalized under Napoléon III), and slight corporatist undertones.
However, that came to an end with the death of the Prince in South Africa. Influenced by his mother and numerous personalities among conservative Bonapartists, the Prince Impérial had developped the same distrust his father had with Prince Napoléon and in his will, he had designated Prince Napoléon's son as his successor. This is where things begun to fall apart for the Bonapartists. Indeed, by passing over Prince Napoléon as head of the Bonaparte family, the Prince Impérial's will created a rift in the Bonapartist movement, blowing up its unity. This first translated into those supporting the dynastic tradition, therefore Prince Napoléon's rights, and those supporting the Prince Impérial's will, therefore Prince Victor's right. No need to say, this was doubled with political leanings of each competing Bonapartist factions. Also, the break was made all the more brutal by the character of Prince Napoléon who saw his own son siding against him and was driven further apart from the movement: the man was a Corsican to the bone if you allow me, one to allow lifetime grudges and who saw the Bonapartes as a family more than a dynasty, and even on his deathbed, he refused to see his son. Though the dynastical dispute would only be solved with Prince Napoléon's death in 1891, the harm was done.
From this point, the different factions of the Bonapartist either left the ship and rallied the Republicans or the Monarchists, or they became increasingly marginalized in the French political scene. Prince Napoléon for instance took position in favor of a Republican model, though a presidential regime with Bonapartist undertones, quite what de Gaulle set up in 1958 actually. Meanwhile, the dominating conservative faction that had taken over the movement further drift rightwards and sought closer bonds with the Monarchist right, but in the process, they lost their visibility by entering this alliance; in the hindsight, without a clear leader to steady the ship, it was more or less unavoidable they would do so. At each passing election, the Appel au Peuple lost more and more seats until after the Boulangist crisis in 1889, they became practically invisible.
If you wish to retain the Bonapartists as a significant political force, capable of an imperial restauration, even after the Great War, in my opinion, you'd need to save the life of the Prince Impérial. As he was born in 1856, he'd still probably be around by the time of the Great War, being only 58 in 1914, 73 by the time of 1929 Crash, and 83 at the beginning of World War 2. Actually, the infamous Philippe Pétain was born barely a month and a half after the Prince Impérial and he died in 1951, aged 95. Even though I think the Bonapartists would probably have a tough time into the 20th century (even the Monarchists were past as a political force past the 1890s and were subsumed into the conservative Republican right), and even through the Boulangist crisis (rumors of an imperial restoration would more than likely to stir up the Republicans' already paranoid stance against them), they would retain key advantages the Royalists didn't have: a clear leader and a united movement.
Because with the Prince Impérial is still around, the Bonapartists wouldn't be bound to merge with the Monarchist bloc into the 1880s and remain a distinct group, if not only for the sake of not alienating the left wing. Meanwhile, the Royalists were bickering over politics and the throne between Legitimist and Orleanist claimants ever since the beginning of the Third Republic, frustrating any serious effort at a restauration while the said claimants were relatively unknowns to the common people, unlike the Bonaparte heir who, being the son of the last Emperor, was a living reminder to everyone about the Second Empire. Incidentally, that would put the Prince Impérial in a very good position for an imperial restauration during the Boulangist crisis in 1889, but that too could well not happen if Boulanger refuses to go for a coup as he did IOTL. Then, unlike Napoléon Victor who rallied lately around his father's presidential republic platform, it's doubtfoul that the Prince Impérial renounces his right to the throne. The next crisis to come, the Dreyfus affaire, could too prove challenging as it provided a strong rallying point to the Republican left after the Panama scandal who eclipsed the right until 1919. However, if even Paul de Cassagnac, a Bonapartist leaning on the far-right, like Paul Déroulède, a nationalist and former boulangist, both came to believe in Dreyfus' innocence, why not the Prince Impérial?
I'd say that in these times, after a downturn in 1889 following the Boulangist crisis, despite a brief upsurge in 1893 elections following the Panama scandal, the Appel au Peuple would be reduced to about one or two dozen deputies into the 1900s, but retain a small presence in the Palais Bourbon even during the Great War, rise again and perhaps double their score in 1919 before falling back again in 1924. All in all, this scenario is only a political development that could have relatively minor repercussions on the whole picture, as one more dynastic claimant would not significantly change French history through the whole period, this keeping butterflies minimal. I'll leave up the rest to you.