Napoleon conquers Spain and Italy, and holds them long enough for societies and languages to merge into one.
Well, that's going to be awfully hard considering that the Spanish heavily resisted Napoleon and resistance against the French would form a basis for the survival of a separate Spanish language and culture no matter what the French try. Italy would be more sympathetic.
It would actually be more likely if Revolutionary France stayed a Republic, with Napoleon maybe still becoming a First Consul but otherwise staying staunchly republican, not trying to found a dynasty and eventually losing power when France has finished the subjugation of its enemies. Trying to unite the Romance languages in the framework of either a single state or a confederation of states (probably the latter, as nationalism will be an issue in case of the former) is much, much easier in a republican framework, thanks to ideological solidarity. A French-led loose confederation of republics (kind of a proto-EU, in fact), if it formed, would need to pretty much achieve most of Napoleon's dreams and hegemonize Central Europe, or the old order would try to destroy them again and again, and eventually succeed.
The republican ideal of "Liberté, Egalité, Fraternité" would also serve to provide justification for the common language (to tighten the bonds of brotherhood). At some point, no later than in the mid-1800s, someone proposes the idea of creating a common Romance language (preferably someone who is not French, as that might have negative connotations of "French cultural imperialism"), and through some miracle, others agree. I think it's going to be pretty damn hard for this language to supplant any of the existing languages, but there might be a "Romance" language that's used as a lingua franca within this hypothetical republican confederation, learnt as a second language by most, and a first language by a few people. It's essentially a conlang that combines the most desirable elements of various languages and would probably also undergo a process of simplification and a few ideological modifications.
This is the closest I can get that could theoretically happen IMO. As for what areas this language could be prevalent in? I'd imagine France, Italy (the first sister-republic), possibly Dalmatia (as part of a Dalmatian cultural revival), maybe Catalonia (as a kind of opposition to Castilian Spanish), the Helvetic Republic... I doubt it can be extended to all of Iberia this way, Spain shares a language with half of the Americas and would probably not want to move away from that.