The continuation of Monstrelet's chronicle (which consists of almost everything in said chronicle after 1444; Monstrelet himself died in 1453), written by Mathieu d'Escouchy, which is in large part cribbing from the Grandes Chronique's de France as well as Jean Chartier has this to say:

". . . the grand Turk with a numerous army of Saracens had invaded Christendom; that he had already conquered the noble of city of Constantinople and almost all Greece; that he had captured the emperor of Greece, had caused caused him to be inhumanly beheaded. . ."

He goes on to accuse the Sultan of various other terrible deeds like burning down Hagia Sophia. All of this should be balanced against the well documented Ottoman policy of trying to win over their conquered subjects and the status afforded to Orthodox Christianity within the empire. Also, it would have been a bit hard for Sultan Mehmet to have turned Hagia Sophia into a mosque had he burned it down.

This account comes from a letter from the Pope to the Duke of Burgundy, who is urged to go on crusade against the Turks. And frankly, these accounts of Turkish atrocities in capturing Constantinople remind me of nothing so much as the various versions of Pope Urban's speech calling the First Crusade at the Council of Claremont.

In a later chapter the siege of Constantinople is actually described (because that's how the continuation of Monstrelet rolls) and the conflict is clearly presented as between Turks and Christians; not Romans. This is surely in part due to the coalition that was defending Constantinople. But there is also certainly an element of this being portrayed as a great clash between the Christian and Muslim worlds. The chronicle here refers to "The emperor of Constantinople".

The continuation does mention Rome, but not in association with the Byzantines:

". . . the grand Turk, when only twenty-tree or twenty-four years old, was more cruel than Nero, and delighted in shedding blood: he was bold and ambitious, and more ardent to conquer the world than Alexander or Caesar."

Interestingly, the comparison with Alexander also appears in Greek sources and appears to have been part of how Mehmet saw himself. Later, as this Chronicler is setting out his grand and definitely plausible plan for a great crusade he says:

". . . the Greeks will be anxious to recover their lands by the sword"

Again, Byzantium is clearly conceived of as a lost Greek empire; there's no association in the chronicle between it and Rome. Where allusions to antiquity are coming up they're in comparing Mehmet to ancient conquerors.

The Chronicler recounts a letter from the Sultan to the Pope. The French used is "desquelles la teneur s'ensuit" which the translation I've been using renders as "The tenour of the last was as follows". I'm not sure whether to interpret this as a claim to providing the literal content of the letter, or just the gist of it - anybody with more knowledge of Middle French would be a big help here. Regardless, this letter, the text of which we should take with an entire shaker full of salt (considering that not only is he highly unlikely to have actually seen correspondence between the Pope and Sultan for himself, but he's absolutely hostile to the Sultan) says:

"We marvel, therefore, and grieve, that the Italians should be our enemies - for we are naturally inclined to be attached to them, as being, like to ourselves, of the issue of the trojan race, and of ancient birth. We are sprung from the same blood, and regularly descended from king Priam and his line. . ."

Again, there's an attempt to locate the Turks within the context of the classical past. Indeed the Aeniad quite famously attempts to connect the Romans to the Trojans.

Another interesting passage:

"We have also the intention of restoring Troy the great, and to avenge the blood of Hector and the queen Ixion, by subjecting to our government the empire of Greece and punishing the descendants of the transgressors."

This is actually pretty definitive in establishing a link between classical Greece and the Byzantine Empire and not classical Rome in the mind of this chronicler. I feel pretty comfortable saying here that the chronicler is putting words in the Sultan's mouth; none of this jives with his actual policy, which was one of substantial leniency towards Orthodox Christians. Nor does it jive with his presentation of himself as 'Kaysar-i-Rum' which sort of doesn't work so well if the thing you just took over was a Greek and not a Roman empire.

The chronicler also denies the Venetians a claim to the Roman past:

"we know the Venetians to be a distinct people, in their manners and laws, from the Romans."

Then he has Mehmet invoke Jupiter and assorted other Roman gods. Again, this is another sign of the chronicler putting words in the Sultan's mouth because I am pretty sure Sultan Mehmet II did not worship "our great god Jupiter". I'd absolutely love somebody more well versed in Ottoman history to weigh in on how much they think this chronicle is carrying echoes of Mehmet's own propaganda. /u/Chamboz if you have the time?

So, let's call this a Franco-Burgundian perspective. The Chronicler himself is a shameless partisan of the Duke of Burgundy and the stuff he cribs from Chartier and the Grandes Chroniques represents the French perspective. He is, clearly, part of a long medieval tradition of identifying the Byzantine Empire as Greek, as the Greek Empire. He shows no knowledge of Constantinople's ties to ancient Rome, looking instead to classical Greece for historical parallel. To the Chronicler the fall of Constantinople represented a major blow to the Christian world. You can see this is just how much space he devotes to it - the continuation is fixated on France in a way that Monstrelet himself was not, so this is quite the exception. But, it does not appear to this chronicler to have represented the fall of the Roman Empire in any meaningful sense.

Jean de Waurin, another Burgundian chronicler, of this period, gives basically identical details. I'll spare you the horrors of my attempts to translate it into English but again, it's described as the conquest of "Constantinople and almost all of Greece". Again, as we saw with Mathieu d'Escouchy the focus is on Constantinople as a Christian city.

Whilst I'm not even going to try and claim that I can speak for the writers of any other region than those that I've mentioned I hope this gives you some idea of how the fall of Constantinople was perceived in French and Burgundian chroniclers. The event was a big enough deal that we could be here all day going through the reactions to it in Christian histories and art.

Famously, Mehmet himself did recognise the connection between Constantinople and Rome and titled himself 'Kayser-i Rum'. But the couple of books on Ottoman history I have don't really go into this, so hopefully somebody else can give you actual detail on the way continuity manifested itself.

The divorce of the Byzantine Empire from its Roman legacy came about fairly early on; centuries before the city fell. A good example of this is in Liudprand of Cremona's history of the Reign of Otto I from the 10th century. Here Liutprand is already referring to the Byzantines as Greeks, for example: "At this same period Simeon of Bulgaria began to press the Greeks very hard." Nevertheless, Liutprand is well aware of the connections between Byzantium and Rome, for example when he's attempting to insult the Emperor Nikephoras he says "For us the word Roman is comprehended every form of lowness". This is shortly after the Emperor has tried to insult Liutprand by pointing out that his people, the Lombards, aren't Romans.

As for the Greek sources, as you might expect, they view the fall of the city as something of a calamity. Furthermore, they're keening aware of the ties between Constantinople and Rome. For example, when Doukas is scolding the citizens of the city he says "You miserable Romans, you wretches". Kritoboulos says that the destruction of the Romans was a huge event. Yet, these sources also make reference to the 'Hellenes' which evokes a classical mold of Greek identity. I really would be so far out of my lane as to just be speculating on how 'Roman' and 'Hellene' as categories of identity interacted in the late Byzantine world, so any help on this one would be hugely appreciated.

Sources:
Enguerrand de Monstrelet, Chronicles, trans. Thomas Johnes, 1810 (my kingdom for a modern critical edition of Monstrelet with facing text).
Jean de Waurin, Chronicles Vol. 5, edited by Sir William Hardy.
Liutprand of Cremona, History of Otto.
M. Nikolić, 'The Greatest Misfortune in the Oikoumene Byzantine Historiography on the Fall of Constantinople in 1453', in Balcanica, vol. 47 (2016), pp. 119-133.
H. Inalcik, Essays in Ottoman History