The exact when, as well as how much the Ottomans conquer, and how long they hold Rome, are incredibly important to the scenario. How long they hold Rome, in turn, is dependant on how capable their rulers are and what their relations with western Europe are like. Since the OP says early 15th century, I'll work from there.
Going with Selim I's reign, which gives us a nice short little window of eight years under which to accomplish it, which IOTL saw rapid expansion of the Ottoman state, it's pretty much not gonna happen. There is no reason for Selim to attack Italy. The Safavids are breathing down his neck in the east, the Mamluks are collapsing to the south, leaving him with plenty of reason to turn east. Italy, on the other hand, is under the control of Spain in the south, which in turn is quite active in the region and strong. The Papacy itself is pretty strong following the reign of Julius II, judging by its ability to wage war on the surrounding Italian states, and France itself is in control of Milan. If the Ottomans get through all of this, then not only are they idiots for even trying in the first place, they are idiots with ASBs on their side. If we wait until later then Charles V comes into his majority and inherits the Spanish throne including Naples, and then he gets the HRE as well. If we go earlier then the Ottomans are going to be weaker. Earlier, and the Ottomans don't have the naval strength for long term projection.
Best shot is probably Mehmed II's invasion, but he died so early in that it's hard to tell how far it was capable of going, and my best estimation says it isn't going to get far. While it is certainly an illusion that there would be an instant eternal crusade, when the Ottomans took Otranto IOTL France, Hungary, and several Italian states responded to the Pope's call to action, with Hungary and the Italians actually providing troops. There is going to be stiff resistance for sure, and the more troops Mehmed sends, the harder it gets to supply them, and the more risk he puts the rest of his empire at.
Basically, Ottoman Italy is a hard sell in and of itself, and without knowing how it comes about we can't make any meaningful insights into what happens when they take Rome.
Going with Selim I's reign, which gives us a nice short little window of eight years under which to accomplish it, which IOTL saw rapid expansion of the Ottoman state, it's pretty much not gonna happen. There is no reason for Selim to attack Italy. The Safavids are breathing down his neck in the east, the Mamluks are collapsing to the south, leaving him with plenty of reason to turn east. Italy, on the other hand, is under the control of Spain in the south, which in turn is quite active in the region and strong. The Papacy itself is pretty strong following the reign of Julius II, judging by its ability to wage war on the surrounding Italian states, and France itself is in control of Milan. If the Ottomans get through all of this, then not only are they idiots for even trying in the first place, they are idiots with ASBs on their side. If we wait until later then Charles V comes into his majority and inherits the Spanish throne including Naples, and then he gets the HRE as well. If we go earlier then the Ottomans are going to be weaker. Earlier, and the Ottomans don't have the naval strength for long term projection.
Best shot is probably Mehmed II's invasion, but he died so early in that it's hard to tell how far it was capable of going, and my best estimation says it isn't going to get far. While it is certainly an illusion that there would be an instant eternal crusade, when the Ottomans took Otranto IOTL France, Hungary, and several Italian states responded to the Pope's call to action, with Hungary and the Italians actually providing troops. There is going to be stiff resistance for sure, and the more troops Mehmed sends, the harder it gets to supply them, and the more risk he puts the rest of his empire at.
Basically, Ottoman Italy is a hard sell in and of itself, and without knowing how it comes about we can't make any meaningful insights into what happens when they take Rome.