What religions did Constantine convert to?

I think it's a bit misleading to suggest that Constantine's conversion was on his death bed. Yes, he did his baptism on his deathbed, but he was a pretty heavy follower well before then.

At the battle of the Milvian bridge where his victory set him as the sole ruler of Rome:

https://www.britannica.com/topic/Battle-of-the-Milvian-Bridge

On 27 October, the night before the battle, it is said that Constantine had a dream: he saw the sun—the object of his own worship—overlain by the figure of a cross. Beneath it was inscribed the simple message in hoc signo vinces, which translates as "In this sign, prevail." Constantine needed no further persuasion. The next morning he ordered his men to paint crosses upon their shields. They then marched into war, accordingly, as "Christian soldiers."

There's a couple parts that suggest his devotion bordered on superstition.

More from Britannica:

https://www.britannica.com/biography/Constantine-I-Roman-emperor#ref384510

Constantine’s chief concern was that a divided church would offend the Christian God and so bring divine vengeance upon the Roman Empire and Constantine himself. Schism, in Constantine’s view, was inspired by Satan.

By 313 he had already donated to the bishop of Rome the imperial property of the Lateran, where a new cathedral, the Basilica Constantiniana (now San Giovanni in Laterano), soon rose. The church of St. Sebastian was also probably begun at this time, and it was in these early years of his reign that Constantine began issuing laws conveying upon the church and its clergy fiscal and legal privileges and immunities from civic burdens. As he said in a letter of 313 to the proconsul of Africa, the Christian clergy should not be distracted by secular offices from their religious duties “…for when they are free to render supreme service to the Divinity, it is evident that they confer great benefit upon the affairs of state.” In another such letter, directed to the bishop of Carthage, Constantine mentioned the Spanish bishop Hosius, who was important later in the reign as his adviser and possibly—since he may well have been with Constantine in Gaul before the campaign against Maxentius—instrumental in the conversion of the emperor.

I'm having problems finding any reference that would suggest that Constantine was anything short of a devoted (to the point of superstition) Christian. It was far more than a simple deathbed repent.

From the comments and to add as a summary:

It was not unusual at this time for a Christian to be unbaptized, to the point where the majority of those identifying as Christian were not baptized. They were known as the catechumen https://www.britannica.com/topic/catechumen. It was somewhat a mix of fear as less than 10 years prior, Diocletian had initiated what became known as the great prosecution and a mix of obscurity (Christianity at this time could be referred to as a 'fringe cult' as it was far from mainstream). Baptism isn't what it is today (from the link above) "The baptismal rites now used are adaptations of rites intended for the reception of adult catechumens.".

With that in mind, all signs shown by Constantine are nothing less than that of a devoted follower of Christianity. He single handily donated a large portion of what would become church property and brought Christianity into the mainstream. There is no evidence that he would begin converting to other religions upon his deathbed.

It is very difficult to actually find any convincing and reliable account of the true religious affiliations, feelings and beliefs of Constantine. His actions were so not in a straight line that one has to conclude that his own convictions were apparently ambiguous or "flexible". It is probably imprecise to call him an unequivocally devout and exclusively Christian believer (a "fundamentalist" in today's terms?) He was a public convert and baptised on his deathbed. But his whole life is much more colourful in its nuances as to allow to call him just a Christian.

Constantine himself is in no small way responsible for creating many of the uncertainties about his religious convictions and religious policies which have been the subject of scholarly controversy since the sixteenth century. He was a highly skilful politician who, like all others of his breed, appreciated the necessity of using deceit in achieving his aims, and he had no compunction about eliminating those who obstructed his dynastic plans (Chapter 5). Moreover, he consistently employed propaganda in order to perpetuate deliberate falsehoods about both himself and important political and dynastic matters. Constantine’s subjects perforce accepted official falsehoods and reiterated them in public – and many no doubt genuinely believed them, as so often happens even in our modern world. Gross falsehoods put out by what may aptly be described as Constantine’s propaganda machine for con- temporary consumption have also deceived many recent historians of Constantine and the Later Roman Empire – even those who prided themselves most on their critical acumen.
(From Timothy Barnes: "Constantine. Dynasty, Religion and Power in the Later Roman Empire", Blackwell: Malden, Oxford, 2014, p2.)

It seems to be the case that he was much more interested in Christianity than would have been expected from a truly pagan contender for the highest office in the Roman Empire. But the universally repeated account of him having a vision of unquestionable Cristian significance (was it a cross or the ChiRho?) that caused an on the spot conversion seems no longer viable. Contemporary accounts from the time of Constantine have been met with scholarly criticism that tries to reconcile facts with tradition and explain the discrepancies. (Peter Weiss: "The vision of Constantine", Journal of Roman Archaeology, Volume 16 2003 , pp. 237-259.)

The fundamental reason is that Weiss showed how the hypothesis that Constantine saw a solar halo in 310, to which he only later gave a Christian interpretation, explains all the early evidence in a way which no earlier hypothesis had ever done.
(From Timothy Barnes: "Constantine. Dynasty, Religion and Power in the Later Roman Empire", Blackwell: Malden, Oxford, 2014, p75.)

After the death of Constantine his own writings, which are not free from manipulative deceptions our primary sources are hagiographic in nature. Not a good basis for reliability.

Not only did Constantine's life fall into place in Eusebius' historical interpretation, but it could become an effective paradigm for others contemplating the same decision, or hoping for the same redemption. This sequence was also seductive because it was an interpretive fiction. Eusebius imposed a coherence in his Lift in order to compensate for the messiness and ambiguity in the emperor's career. The most daunting obstacle to seeing Constantine's life in terms of one sudden transition and a subsequent consistency is, in fact, his own life. (p134.)

But there are some circumstantial pieces of evidence that allow a glimpse of the complicated realities in the still mixed and tolerant empire. Among them is a giant one:

In the end, the statue did represent a conversion experience. But it was not the conversion preferred by Eusebius, or by modern accounts. Popular gossip claimed that the statue had once depicted Apollo in his guise as Helios the sun god, until it had been reworked into an image of Constantine. The conversion of the statue mimicked the transformation of Constantine, who had early on been an adherent of Apollo. But after the statue had been converted, people did not refer to it as "Constantine" or "Christian emperor;" Instead, they called it Anelios. Apparently it was easier to think of the emperor in terms of what he had left behind, rather than in terms of what he had become. This statue was, simply, "Not the Sun." (From Raymond Van Dam: "The Many Conversions of the Emperor Constantine", in: Kenneth Mills & Anthony Grafton (Eds): "Conversion in Late Antiquity and the Early Middle Ages. Seeing and Believing", University of Rochester Press: Rochester, 2003, p 147.)

Constantine indeed did cover his bases. He did so in the most traditional way imaginable for a pagan emperor. That does not include converting to Judaism. But it does include keeping the old faith (whether for himself, for the empire, or any combination thereof):

Constantine retained this connection with Apollo for years. A gold medallion minted in early 313 depicted the emperor in profile with Apollo, who was wearing a solar crown, and the sun god Sol remained on Constantine's coins for over a decade. Sol also appeared in a medallion on Constantine's triumphal arch at Rome, neady correlated with a frieze depicting the emperor's arrival at Rome in 312. Nor did the emperor terminate his support for pagan priests and practices after he began to patronize Christianity. In 320 he allowed the consultation of soothsayers when buildings were struck by lightning, and after 324 he extended his support to a pagan priest. In the mid-330s he was still permitting the construction of a new temple in Italy. Constantine's vision of Apollo should qualify as a conversion experience. (p135.)

Remember that the traditional story follows Eusebius's account of complete and thorough conversion, suddenly and absolutely, in 312! The archaeological reality of the evidence found is just not compatible with this pious narrative.

In contrast, Constantine himself apparently liked to tell stories about his many visions and conversion experiences. Consideration of only these three episodes suggests that Constantine may have had many conversion moments, and that his religious beliefs throughout his reign were not as consistent as Eusebius presented them. Despite his evident patronage of Christianity, his life included changes of mind, uncertainties, contradictions, and ambiguities. In other words, it was anormal life. (p 137.)

Apparently he requested that a giant statue of himself at Rome should hold a cross in its right hand, and that the dedicatory inscription should commemorate his devotion to the Savior: "I have liberated your city by this sign of salvation." In contrast, the dedicatory inscription on the huge triumphal arch completed at Rome in 315 was much more bland and noncommittal. It attributed Constantine's success merely to "the impulse of a divinity." (p136)

And this base-covering flexibility and – let's call it open-mindedness continued through to the very end of his life.

Constantine spent the last six days of his life at Achyron, an imperial estate a little way from Nicomedia. As he felt death approaching, Eusebius of Nicomedia baptized him. No members of his immediate family were present, but he was surrounded by the vast extended family with which he had spent so much of his life, the imperial court. It is hard to imagine that Evagrius was not in the room when he died; and the presence of both the Christian bishop and the pagan prefect at his deathbed—if they were both there—expressed Constantine’s dual aim during the last several decades of his life: namely, to worship the god who brought him victory and to fulfil his destiny as emperor in a way to equal, even exceed, the achievements of those who had gone before him.
Around noon, on May 22, 337, Constantine breathed his last.
(From: David Potter: "Constantine the Emperor", Oxford University Press: Oxford, New York, 2013, p 313.)

That he was baptised as one of the last things he went through in life seems uncontroversial. Whether he was old Roman pagan, Mithraist, Arian, Nicean creed Christian in terms of true belief seems hard to ascertain for any one point in time and even more uncertain for a stringent line of his life. Instead, one thing seems very certain, that he was a true politician.