Short answer:

There were instances of reprisals, generally referred to as "The [Second] White Terror of 1815" but they were limited in time and place, and the circumstances were very different than after WW II. You do not have the kind of brutalization of women or collaborators that you see in 1945. The restored Bourbons are more vengeful to sometime Revolutionaries who'd voted for the execution of Louis XVI than they are to supporters of Napoleon.

Discussion:

The situation "after Napoleon's defeat" is very different than the situation after the Second World War. Start with the most basic historical detail: Napoleon is "defeated" not once, but twice. He is defeated and abdicates in 1814 and is exiled in to Elba. In 1815, he escapes Elba and launches the 100 Days campaign, ending with his second defeat at Waterloo, and his final exile to St. Helena. So you have "the defeat of 1814" and "the defeat of 1815", an important distinction from 1945.

As a result of these two distinct defeats, you have distinct Bourbon Restoration regimes, and distinctions in those who'd served the Napoleonic regime: there were those who pledged loyalty to the Bourbons in 1814 -- and remained loyal when Napoleon escapes from Elba.

You have a second category -- those who swore allegiance to the Bourbons in 1814, but then rallied to Napoleon in 1815. These men had more problems-- the most famous of them would be Marshal Ney, who is tried for treason and executed in December 1815. But you can compare with Marshal Soult, who also is with Napoleon at Waterloo. He's in disfavor for a time, but is recalled by Louis XVIII in 1819, made a Marshal of the restored French monarchy. What accounts for the difference between his fate and Ney's? Probably that Ney was simply the first one to go on trial, and the trial was deeply unpopular. Having executed him, other figures did a bit of grovelling and Louis XVIII and his allies thought better of more trials.

As an example of the complexities of loyalty during this period, consider Napoleon's sometime Foreign Minister, Talleyrand. He plays a major part in engineering the return of the Bourbons, and serves as Louis XVIII's Foreign Minister, and the French representative at the Congress of Vienna. His illegitimate son, the Comte de Flahaut, rallies to Napoleon in 1815 . . . and after Waterloo flees, likely of necessity, to England. Talleyrand is pushed out of power in 1816 by the rising ultras-- not subject to any vengeance, but the Comte d'Artois doesn't much care for him. Talleyrand returns the favor, and in 1830 is part of the July Revolution which disposes of the Comte (now Charles X); note that the restored Bourbons only last 15 years in power . . . Louis XVIII recognized the fragility of his position, Charles X didn't.

When Louis XVIII returns in 1814, he's largely forgiving, when he returns in 1815, he's still substantially forgiving, but the "ultra-royalist" Bourbon supporters are in the ascendency, and by 1816, you see some more vindictive measures but their greatest venom wasn't necessarily directed at supporters of Napoleon. The prevailing suspicion was more against revolutionaries, men who might have been associated with the execution of the late King Louis XVI and his family. Even then, it is extraordinary to see that some of the regicides, like Joseph Fouché, serve the Bourbons for a time after their return as their Police Minister. Initially he draws up lists of other regicides who are exiled, and then he himself is exiled. It's a purge, but not a particularly bloody one. So by the end of 1816, most of the sometime revolutionary zealots have lost their jobs and many are exiled, but not many are pursued further.

Talleyrand had created a convenient historical fiction for France -- and a convenient one for himself. France had been ruled by a "usurper", and men like him had continued to serve France loyally; now the legitimate government had returned, they could happily continue to serve France. In most places, this pragmatic solution worked best, you could draw the analogy to Austria's position on the the years of 1938-45; "we were just victims of the usurper".

French domestic administration during the Napoleonic years was substantially directed by the Prefects, and in 1814 the returning Bourbons kept them on. The Comte d'Artois as Lieutenant-General sent a group of "Commissaires-Extraordinaires" to establish contact with these prefects, who were kept in place. Indeed, France essentially never undid the Napoleonic system of administration, which to a substantial extent remains in place to this day.

Some of the greatest violence in the aftermath of the Bourbons' return in 1815 is in the Languedoc, and while its referred to as the [Second] "White Terror"; it has roots in religious tensions that are essentially unrelated to Napoleon or the Bourbons-- it's about the affiliation of urban protestant merchants with the Revolution, and continuing clashes with more agricultural catholics dating back to the 1790s and indeed much earlier.

The two leading terrorists in the Gard, responsible for the great majority of crimes committed in 1815, were both agricultural workers. Jacques Dupont, known to his accomplices and victims alike as Trestaillons (supposedly after his practice of cutting his victims into three pieces), was a travailleur de terre "completely uneducated and without any means of subsistence". It was alleged that the personal motivation behind his terrorist activities was the fact that "his father had been killed by Protestants during the bagarre de Nimes along with nine hundred other catholics". It is interesting that Vidocq, the arch-criminal who became one of the founders of the Surêté nationale, wrote in his colourful memoirs that Trestaillons was the last of the chauffeurs, the brigands of the ancien regime who occasionally laid out their victims on burning beds of straw. At least one of the victims of the White Terror in 1815 was burned alive. In Uzès, Trestaillons's crimes were duplicated by Jacques Graffand (in honour of his mentor he called himself Quatretaillons!). Graffand had also risen from the ranks of the unemployed. His devotion to the royalist cause provoked him into one last desperate gamble on behalf of the Bourbons when, after the July Revolution of 1830, he led an abortive counter-revolution in the Gard on behalf of Charles X.

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Sources:

Payne, Howard C. “The Bourbon Restoration's Commissaires-Extraordinaires Du Roi in 1814.” French Historical Studies, vol. 9, no. 1, 1975, pp. 37–62.

Kieswetter, James K. “The Imperial Restoration: Continuity in Personnel and Policy under Napoleon I and Louis XVIII.” The Historian, vol. 45, no. 1, 1982, pp. 31–46.

DE RICAUMONT, JACQUES. “LA TERREUR BLANCHE.” La Nouvelle Revue Des Deux Mondes, 1981, pp. 363–367.

Lewis, Gwynn. “The White Terror of 1815 in the Department of the Gard: Counter-Revolution, Continuity and the Individual.” Past & Present, no. 58, 1973, pp. 108–135.

Newman, Edgar Leon. “LOST ILLUSIONS: THE REGICIDES IN FRANCE DURING THE BOURBON RESTORATION.” Nineteenth-Century French Studies, vol. 10, no. 1/2, 1981, pp. 45–74.

Wrede, Martin, and Christine Weil. “Le Portrait Du Roi Restauré, Ou La Fabrication De Louis XVIII.” Revue D'histoire Moderne Et Contemporaine (1954-), vol. 53, no. 2, 2006, pp. 112–138.

G. de Bertier de Sauvigny. “The Bourbon Restoration: One Century of French Historiography.” French Historical Studies, vol. 12, no. 1, 1981, pp. 41–67.

See a good comment on the politics of the Bourbon restoration by u/kieslowskifan here:

https://www.reddit.com/r/AskHistorians/comments/36wsn3/what_is_meant_by_this_quote_about_king_louis/