What happened to the cross on the dome of Hagia Sophia?

Short answer:
Cross? What cross? Was there really a noteworthy one? That seems quite doubtful.

More detailed answer:

A first thought is of course that the depictions shown so far are not entirely realistic. Icons have a halo, real people do not. Churches are marked in pictures with a cross, they do seldom have one in reality, very seldom if it should be that big as the dimensional relations are anywhere near what's depicted.

^{(From a Christian revisionist's imagination, removing minarets and adding a cross, not only golden and giant but also clearly a Latin one, not in line with either Greek or Orthodox styles; click to enlarge)}

The cross on top of the dome of Hagia Sophia presents a further challenge. From an engineering standpoint, it most probably could not have been a massive gold object. Gilded perhaps, but then much lighter. The static on the vault is already a balancing act of empirical trial and error. Adding such a massive weight and wind load is just 'not a good idea'.

In its current structure, the dome of Hagia Sophia consists of four segments created independently of each other in terms of time. A one-piece and continuously constructed pendant dome is therefore to be contrasted with the multi-part dome structure of Hagia Sophia, which was built in three independent construction phases. The assessment of its load-bearing behaviour requires on the one hand the consideration of the static interaction of the four dome segments, on the other hand - within the framework of an analysis of its static-constructive development history - the influences of the partial collapses and the associated intermediate states of the load flow must be taken into account.

^{Fig. 8.6: (a) Original one-piece dome (b) Today's multi-piece dome (From Duppel)}

Looking through for example Prokopios' description of the church De Aedificiis, we find much praise for all the signs of crosses, throughout the church. And one especially big one protecting the city, in the dome. As far as I read it meaning the interior decoration. Not a giant, colossal piece of metal on top of the dome to be seen from afar. One caveat is to be observed in the fact that the lines immediately following this description seem lost now.

A stone cornice, formed on all sides into a beautiful circle, holds all the arched ridges together, on which the hemispherical dome rests and which is touched by the apexes of the arches all around. The artists have placed this cornice on the upper end of the apses as a wreath. On protruding ornaments (485), exposed stones form a narrow path on which the man who provides the lights can walk around safely and light the holy lamps. Above this, however, the dome rises up into the boundless space and curves on all sides to form a sphere, like the light sky of the house encloses the roof (490).

At the apex, however, art formed the sign of the cross to protect the city. It is even astonishing to see how gradually the dome rises, further down and increasingly narrower up. However, it does not end in a sharp point, but rather (495) gives the impression of a sphere floating in the air. The artist placed the circular base on the well-jointed curves of the arches ... with the hands of the stonemasons he alternately smoothed the rising ribs. At the sight of them one could think they were a ring-shaped comb, to which nature (500) [... Text corrupted until 505]
_{(Caveat the translation and interpretation. After all, it's all Greek to me.)}

The church was a Christian church, as such not like pagan temples to be primarily working as an a nice to look at from outside building, one into which only priests and the gods should enter. With Christian places the action is where the community is: inside. The exterior is much less important, although often as quite magnificent.

As with Hagia Sophia itself:

But if the outside inspires no strong feeling of admiration one has only to pass the threshold to realize the genius of the designers.
_{— Thomas Graham Jackson: "Byzantine and Romanesque Architecture", Cambridge University Press: Cambridge, 1913.
(Not mentioning any cross atop the dome, but showing nicely how various descriptions of the architectural process were often flawed from the beginning and then later on copied with adding ever more errors to them.)}

Also note that Prokopios describes the first version of the Justinian building, before the first dome collapsed. Just adding some more weight to the argument, that we probably will not find much of the whereabouts of a perhaps imaginary cross. It quite certainly was not a giant piece of metal.

These considerations aside, some rather shady sources claim for example that 'Ismael Hamze the city commander ordered 'the cross' being taken down.' And that it was re-purposed as a humiliation: converted into a staircase for the mosque. Note that these tidbits are copy-pasted countless times around the web, without proper sources for these assertions. An evangelical pastor even preached to his sheep that the giant cross fell down into the streets, plunging into the knee deep blood of the faithful spilled by the infidel invaders…
_{(not linking that)}

One simple question for plausibility: did the Ottomans take a giant all-metal golden cross down from a delicately stable dome to remelt it into an entrance staircase so that visitors to the mosque would trample with their feet upon the symbol of Christianity?

The 'Fall of Constantinople' was certainly not 'a nice event', but many details of the descriptions were quite fanciful nonetheless. In many cases clearly just inventing stuff for propagandistic reasons and imagination that is most unlikely.
_{(— Marios Philipides & Walter K. Hanak: "The Siege and Fall of Constantinople in 1453. Historiography, Topography, and Military Studies", Ashgate: Farnham, Burlington, 2011.)}

Those sources I checked seem to hardly ever mention any cross, let alone a big metal one, on top of the dome.

If there should have been a golden or gilded cross to be noteworthy, then it seems to be quite more plausible that it was gone by the time the Turks entered the city. The Sack of Constantinople in already in 1204 is a more likely date for such an event:

Despite their oaths and the threat of excommunication, the Crusaders systematically violated the city's holy sanctuaries, destroying or stealing all they could lay hands on; nothing was spared, not even the tombs of the emperors inside the St Apostles church. The civilian population of Constantinople were subject to the Crusaders' ruthless lust for spoils and glory; thousands of them were killed in cold blood. Women, including nuns, were raped by the Crusader army, which also sacked churches, monasteries and convents. The very altars of these churches were smashed and torn to pieces for their gold and marble by the warriors. Although the Venetians engaged in looting too, their actions were far more restrained. Doge Dandolo still appeared to have far more control over his men. Rather than wantonly destroying all around like their comrades, the Venetians stole religious relics and works of art, which they would later take to Venice to adorn their own churches.

Also note that while some modern reconstructions sometimes do insert various forms of a cross on top, for example the one for the adjacent Augustaion, intact up to 1204 already does not:

If there was one cross, then the timeline for this possibility is quite restricted:

[…] Ottoman Sultan Mehmed II (r. 1444–46; 1451–81) ordered the church’s conversion into a mosque. […] the bell, relics, crosses, icons, and the cross on top of the dome were removed; […]
_{— Gábor Ágoston & Bruce Masters: "Encyclopedia of the Ottoman empire", Facts on File: New York, 2009, p243.}

But the size of it surely overstated:

The conqueror supplanted the large metal cross that crowned the summit of the dome with a crescent. This symbol was replaced in the sixteenth century with a larger and more handsome crescent of bronze, said to have been gilded with 50,000 pieces of gold and to be visible 100 miles out to sea.
_{— Richard Winston: "Hagia Sophia. A History", New Word City, 2017.}

These numbers are just impossible. Equally sweeping and in parts contradictory is the following description, again without proper sources:

The first dome was covered with plain gold mosaic, while the new dome of 564 carried a huge cross.^(p158)
_{— Richard Krautheimer: "Early Christian and Byzantine Architecture", Pelican History of Art, Penguin Books: New York, 1964.}

Almost predictably the Turkish Wikipedia page does mention quite a couple of alterations to the building and is not withholding info on Mehmed's orders to convert the church into a mosque. But while crosses from or on the interior are mentioned as removed, this WP variant only knows of structural alterations after the first dome collapse to be made of lighter material, still reinforced in various ways, but never a cross removed from the top.

Quite in contrast many Christian stories and their narratives running wild with what Mehmed did on conquering the Turkish side of this story reads very differently. What Christian sources of the time depict as 'raping a mild virgin on the altar in front of the public' is called 'the sultan making pious devotions at the altar' Regarding the alterations to the church:

Accompanied by a few viziers and courtiers, Mehmed II, from now on known as the Conqueror (Fatih), entered the city that afternoon on horseback. He went first to Hagia Sophia. On entering, his first glance fell on a Turk who, in a destructive frenzy, was hacking away at the marble floor with an ax. The sultan asked him why he was destroying the floor. "For the faith," the Turk replied. Enraged at such barbarism, Mehmed struck at him with his sword and cried, "Content yourselves with the loot and the prisoners. The buildings belong to me." Thereupon the half-dead ruffian was hauled away by the feet and tossed outside. […]

The Conqueror gave no heed to the prohibition of images, as is amply demonstrated by his fondness for the Italian paintings with which he adorned the interior of his palace. As we have seen, none of these works have survived, because Bayezid II on his accession either destroyed them […]

There can be no doubt that superstitious considerations contributed to saving the mysterious monument from destruction. The sultan also gave orders for the equestrian statue of Justinian surmounting a high column in the square known as the Augustaion to be carefully removed, no doubt to save it from the religious frenzy of his troops. These two incidents demonstrate the Conqueror's freedom from religious prejudice, even in his younger years. A further indication is that immediately after the conquest of Constantinople, when the paintings of Hagia Sophia were being whitewashed, he caused the mosaic of Mary in the half-dome of the choir apse to be spared. Toward the end of his life he was totally indifferent to such matters as the prohibition of images. It was not without reason that his pious son branded his liberalism as godlessness.
_{— Franz Babinger: "Mehmed der Eroberer und seine Zeit: Weltenstiirmer einer Zeitenwende", F. Bruckmann: München KG., ²1959}

Note that the Met museum talks of a cross ("Initially, the upper part of the building was minimally decorated in gold with a huge cross in a medallion at the summit of the dome."), but references an almost contemporary woodcut by/after Pieter Coecke van Aelst: "Procession of Sultan Süleyman through the Atmeidan from the frieze Ces Moeurs et fachons de faire de Turcz"

^{The buildings around the Hippodrome have been condensed and slightly rearranged to fit the image, but many are identifiable. Among the monuments of Byzantium is the Church of Hagia Sophia on the right horizon; it is visible just to the left of the Egyptian obelisk erected in the Hippodrome by Theodosius (r. 379–95). The column with the twisted serpents erected by Constantine (r. 306–37) is on the right just behind the sultan. The large tower to the left of Hagia Sophia may be the Galata Tower constructed by the Genoese. Van Aelst traveled to Constantinople in 1533…}

Compared to the Hartmann Schedel Liber Chronicarum for 1453 and of 1493 (WP seems a bit wobbly on dating this?), we see this Sophia:

No cross, no crescent moon yet either… Although other churches have on on this picture and examples like Hagia Irene seem to fall into that category as well.

A scholarly work (Victoria M. Villano: "A Spectacle of Great Beauty: The Changing Faces of Hagia Sophia", Master Thesis, 2012, PDF) referencing the "huge metal cross" uses the above picture which doesn't have it. Equally a city map by Cristoforo Buondelmonti is used in that thesis, and it shows Hagia Sophia in 1422 (variant, even with ocular hole at the top), without a cross on the dome…

A variant of Buondelmonti’s map from 1475 shows a cross, though in a truly gigantic size, almost 3/4th the width of the entire dome:

And the different editions based on this map give really an interesting insight into the development and even evolution of portrayals, or shall we say embellishments:

— Ian R. Manners: "Constructing the Image of a City: The Representation of Constantinopole in Christopher Buondelmonti's Liber Insularum Archipelagi", Annals of the Association of American Geographers, Vol. 87, No. 1 (Mar., 1997), pp. 72-102.

Namely that the further away we get from an actual first drawing the cross on the church gets ever bigger.

_{Notabene from the same source as the first picture by OP: Assassination of emperor Romanos III Argyros, Vatican Manasses manuscript (Vat. Slav. 2), fol. 188v. Reproduced from Filov, Les Miniatures.}

_{View of Constantinople. Notitia dignitatum, Bodleian Library, Ms. Canon. Misc. 378, fol. 84r.}

Both from — Elena N. Boeck: "Constantinople: story spaces or storied imperial places", in: "Imagining the Byzantine Past The Perception of History in the Illustrated Manuscripts of Skylitzes and Manasses".

Again, from the Ottoman side a painting of Hagia Sophia when freshly under Turkish control shows us this drawing, without cross or crescent, but already with two minarettes:

^{Painting of the Hagia Sophia Mosque in Istanbul, illustration in manuscript "Sehname-i Selim Han" (Topkapi Palace Museum Library, A. 3595, fol. 156r).}
_{– Gülru Necipoğlu-Kafadar: "Plans and Models in 15th- and 16th-Century Ottoman Architectural Practice", Journal of the Society of Architectural Historians, Vol. 45, No. 3 (Sep., 1986), pp. 224-243}

When the city was conquered, the church was already in quite a sorry state. Plundered multiple times by besieged Byzantine officials in financial trouble, melting down gold from the church's decoration into coins before. Then from Venetians and other Latins long before the Ottomans had a chance to do much 'damage' directly, plundering with witnesses and evidence now in their churches, "everything they could lay their hands on".
That a cross might have adorned the dome at a time at all is already quite doubtful, as many historical descriptions are not overly reliable, most do not mention one, and drawings, sketches or paintings equally fail to consistently show one, when they otherwise do exactly that for buildings of that kind. In any case, even if there was such a cross at the top, and not just a later embellished invention or fairy tale, it seems quite unlikely that it was all of this: a cross, on the dome, huge, and shiny solid gold, remaining there when Mehmed came and only taken down at his orders, to then be desecrated and used for further humiliation. In fact, most of these isolated and badly referenced, if at all, assertions appear to be awfully in contradiction to the established record.
Sources in a variety of languages fail to mention this as of any significance. Unless convincing evidence comes along that proves the contrary: nothing happened to no cross that wasn't there?

_{— Nevra Necipoglu: "Byzantium Between the Ottomans and the Latins. Politics and Society in the Late Empire", Cambridge University Press: Cambridge, New York, 2009, esp. "Bayezid I’s siege of Constantinople (1394–1402)" (p154).
— Cecil Stewart: "Early Christian, Byzantine, and Romanesque Architecture", Simpson's History of Architectural Development Vol II, Longman's, Green & Co: London, New York, 1954. (No cross for Sophia but careful to locate and depict those roof ornaments when they are confirmed, like for church of Saint Luke of Stiris, in Phocis, with a tiny cross on a vaulted dome…)
— Nadine Schibille: "Hagia Sophia and the Byzantine Aesthetic Experience", Ashgate: Farnham, Burlington, 2014
— Christoph Duppel: "Ingenieurwissenschaftliche Untersuchungen an der Hauptkuppel und den Hauptpfeilern der Hagia Sophia in Istanbul", KIT Scientific Publishing: Karlsruhe, 2010.
— Paul Magdalino: "Studies on the History and Topography of Byzantine Constantinople", Ashgate: Farnham, Burlington, 2007.
— Nevra Necipoğlu (Ed): "Byzantine Constantinople: Monuments, Topography and Everyday Life", The Medieval Mediterranean 33, Brill: Leiden, Boston, 2001.}