So Mehmed II doesn't die, but lands in Italy with his vast army. He defeats the army Naples had sent, and advances on and seizes Naples itself.
Now Rome is in chaos as the Papacy quickly weighs its options. Within a fortnight of Naples' fall, the Pope and his curia have fled the city with as much of the Vatican's wealth as they can fit in the wagons.
The Ottoman army is advancing north, and there is no significant force to prevent them from sacking Rome. Naples' royal family has fled to Spain, to court their powerful cousin King Ferdinand, while the Pope has fled to France ostensibly to raise a new Crusade. The French and Hapsburg are on the edge of war over the Burgundian Inheritance, but the invasion has butterflied away Mary of Burgundy's unfortunate death, so there is not an immediate cause belli between the two powers.
The Pope is preaching a new crusade, and clearly the Ottoman invasion of Italy is a threat to all three European Great Powers: newly united Spain- where the invasion threaten Aragon's Sicily, the French- who have vague claims on Naples and Milan, and the Hapsburg- who don't want the Ottoman's to control their entire southern frontier.
Florence and Milan now face an interesting situation. Ottoman representatives arrive ahead of the Ottoman army, offering peace and the Sultan's recognition of the two cities territory and independence, in return for promises not to host foreign troops. The representatives sweeten the deal by offering to recognize the two cities various territorial ambitions. Venice watches the moves apprehensively, sending messages to its own ambassadors to make sure the Sultan remembers to keep the peace he made with the Serene Republic.
As Lorenzo and Ludivico equivocate over the terms, the Ottomans savagely sack Rome and put their calvary into the field, raiding across the Italian countryside. Help from the beyond the Alps seems to be coming far too slowly. Louis XI raises a large army, but when the Pope dies he seems more interested in using it to help the curia pick the next Pope than fight the Ottoman. Maximilian is reluctant to committ troops while France is raising such a large army and his father, Frederick III, is unable to send troops because the Hungarian King, Matthias Corvinus, has taken the Ottoman invasion of Italy as an opportunity to strengthen his position and expand his control of Christian territory, namely through war with Frederick.
Florence and Milan make peace with the Ottomans. The newly elected Pope Julius II (yes THAT Julius II) excommunicates the Sforza and Medici, and declares that unless their people overthrow them, the entire Florentine Republic and Duchy of Milan will face excommunication. The Italian princes who have made peace with the Ottomans respond by having the "Council of Florence" where a new college of cardinals elects a new Pope. Funds are raised, including significant contributions from the Sultan, to rebuild the Vatican for the new Pope.
In 1483 Louis XI dies, setting off OTL's "Mad War"- a civil war in France over the regency of young Charles VIII, between the Duke of Orelans, the young king's heir, and the Duke of Bourbon, the young king's brother-in-law.
The civil war in France allows Maximilian to make a lasting peace with France, and turn his attention south. His father, the Emperor Frederick, strikes a peace deal with King Matthias of Hungary, recognizing Matthias' territorial gains and recognizing Matthias' bastard son, Janos Corvinus, as Hungary's crown prince. Maximilian has been in contact with Ludivico Sforza, who professes a desire to overthow Ottoman rule. Ludivico and Maximilian make a secret alliance, and King Matthias is brought into it with a marriage agreement between Ludivicio's neice Bianca and Matthias' heir Janos.
So in 1485, Ludivico Sforza welcomes Hapsburg and Hungarian armies into Italy, truly beginning the first round of the "Italian Wars" that would last well into the next century.
I think I might do a timeline.