An important point of Napoléon's life, prior to his ascension as general, is his extensive studies. Between ages 9 and 21, he spend almost all his time in schools (Brienne, Paris, Auxonne), who greatly influenced his personality. For the lieutenant di Buonaparte to be Napoléon, an equivalent formation is necessary.
When the French bought Corsica from the Genovese, it was in a virtual independent state. The Bonaparte/Buonaparte were an important local family linked to Paoli, so in a non-French scenario, Corsica would have gone independent and the Bonaparte would still be on the winning side. The Corsican independence will be backed by at least a great power, be it French or Britain or even Spain. The sons of the new corsican elite will probably be sent to foreign schools in order to strengthen the links between Corsica and its protector, but also in order to gain competent administrators/military officers. A Woolwich-educated Napoléon, maybe ?
For the French not to go into Corsica, you would have to get rid of Choiseul two years prior to OTL (1768 instead of 1770). The "home first" policy of Maupeou-Terray-d'Aiguillon would lead to few foreign interventions. However, the conservative leanings of the triumvirate would probably not lead to a backing of the Corsicans, so these ones will go to the British for support. So, we have an independent, British-dependent Corsica by 1775.
In Corsica, the authoritative side of Pascal Paoli would probably go unchecked, as in OTL. It is quite possible to see Charles Bonaparte having a fallout with his former protector on some subject by 1785. With Charles out, Napoleon will have no further funding for his studies or perspective of career in Corsica. A logic reaction of the Bonaparte clan would be to go to France, the main rival of the British in the Med. So Bonaparte could maybe finish his studies/start his military life in France but as a foreigner, not a Frenchman.
The Corsica annexation had few consequences on the French Revolution, which is likely to happen in this TL in 1789. In these years of Revolution, young exiled Bonaparte will probably see the many opportunities opened by the radical shift in French statehood. Like the exiled Belgians, he could be a rising star in the liberal exiled italian community - the exiled Corsicans being probably too few to be an independent group. As with the Belgians, he will recruit and lead italian liberal units during the expansions phases of the Revolution. His ascension will be as swift as in OTL, but his objective different : from his formative years, he would not see himself as the providential man for France, but for Italy. Starting with Corsica, he will create an Italian Republic, dependent of the French revolutionary government, but his own importance will grow and grow until a breaking point with the French. If the French play ball and let him be the master in Italy (a quite difficult task, to get rid of all internal opposition and to keep the Austrians in check), we could see a true alliance between the two republics, something many in the liberal wing of the Revolutionaries (Talleyrand) would welcome. If not, the British may support him in order to force France to surrender the Netherlands. If he keeps his ambitions to Italy, annoying the Spanish and the Austrians will probably not be enough for France or GB to start a complicated assault on Italy. Maybe in his latter years, a second, more liberal revolution will be the downfall of Consul Napoleone Buonaparte, Founder of the Italian Republic.