My understanding is that the native language of Savoy is neither French nor Occitan but
Franco-Provençal, a collection of related dialects predominating in east-central France and western Switzerland. It is true that speakers of Franco-Provençal do not seem to have identified themselves as speaking regional dialects of this language, of Franco-Provençal as such existing; they defaulted to French. The only remaining territory where Franco-Provençal has not been mostly replaced by France is the Italian
Aosta Valley, where Franco-Provençal persists as a vernacular alongside French and Italian as official languages.
The traditional language of the
County of Nice was a local dialect of Occitan, but Italian was apparently widely spoken. Had it remained Italian, probably Nice would have become an Italophone city with Occitan persisting further away, if at all.
Can Italy keep Savoy and Nice and gain Corsica? Much depends on how Italy unifies, frankly. If Italian unification does not get the support of France, then somehow Italy will need to gain patronage from some country, somewhere. What could this country be? I doubt it would be an Austria deeply invested in the status quo, for one.
One distant possibility might be an Italy that remains closely allied with Prussia and Germany after 1866, to the point of joining Prussia as an ally in the Franco-Prussian War in the hopes of regaining "Italian" territories. Questions about the military capacity of 1870 Italy aside, the country's interests were focused on regaining Rome, a city and a territory much more central to Italian interests than a border city ceded a decade ago and an island lost a century ago and mountain regions never populated by anyone who spoke Italian.