IOTL he did in fact enter politics late in his life, being conscripted as a coast-guard during the Franco-Prussian War, and later becoming one town councillor of Amiens, holding that position until just two years before his death. So what if he enter politics earlier? Verne was initially more of a social commentator than anything. His posthumously published Paris in the Twentieth Century was a scathing critique of French society, economics, and morality in the period, and it was for that very reason that Verne's publisher, Pierre-Jules Hetzel, talked Verne out of publishing it, and the manuscript sat in his vault until 1994. But WI Verne can't be persuaded against publishing his work, and the novel beats Hetzel's expectations and turns out to be another Verne hit.
He continues writing as per IOTL, but his works now have more of a sociopolitical slant to them. Captain Nemo's philosophy is better outlined in ITTL's Twenty Thousand Leagues Under the Sea; The Adventures of Three Englishmen and Three Russians in South Africa spends even more time discussing nationalism and geopolitical/social theory; Around the World in Eighty Days is more overtly a (negative) commentary on the British Empire (and the Anglo-Saxon world in general); The Begum's Fortune is even more a Francophile nationalist tract; Frritt-Flacc becomes Verne's A Christmas Carol; The Vanished Diamond is a critique of European (mostly British) African colonial attitudes; Mathias Sandorf is a political thriller and commentary on the German Question after the Austro-Prussian, Franco-Prussian, and Italian Independence Wars; Two Years' Vacation is Verne's Lord of the Flies; etc.
Largely riding on the popularity of his stories, and his well-established place within the French Liberal tradition, Verne enters government immediately after the Franco-Prussian War, being sent to the new National Assembly in Paris after the fall of the commune, where he quickly stakes out his reputation as a fierce debater and powerful orator. By 1876 his fame is such that he becomes a minister without portfolio under the government of Jules Simon, though the government still falls the next year as per OTL. Verne however returns to high office in 1877 under Jules Armand Dufaure, this time as Interior Minister (replacing OTL's Charles de Freycinet), and from 1877-79 Verne is responsible for an ambitious plan to modernize the country that includes state acquisition of, and expansion of, several hundred kilometers of railways and canals (as per OTL), as well as telegraph wires and what we might call highways (more properly just better standards for major roads).
I see Verne joining the Opportunist Republicans over the Left/Radical Republicans, so by the time 1881 election Verne is well positioned to become Prime Minister under Jules Grévy (once again replacing Charles de Freycinet), where the first thing he does is join the British in bombarding Alexandria during the height of the Urabi Revolt (which IOTL lead to the Anglo-Egyptian War and British conquest). So what happens next?