I broadly agree with those like LSCatilina who think that a late roman empire monotheistic development based on Mithras, Sol, Serapis, etc. is extremely unlikely. Those deities although relatively "new" were already established brand names. How are you going to convince your own devotees (not to mention those of other cults) that they should suddenly start to deny the existence of other gods while you have been perfectly happy (and even eager) to acknowledge their existence for 100 years or more ? The problem with the syncretistic approach shared by all these late-roman religious currents is that, while it is very efficient at building a sizable devotee base in a relatively short time, it precludes you from making the truly fresh start that monotheism requires. If you have been attracting devotees with claims such as "Sol is like Apollo" or "Serapis is like Zeus + Apis", etc. you are stuck with that baggage for ever. You cannot thereafter claim that Appollo, Zeus and Appis "do not exist".
The transition from polytheism to monotheism is only possible among very small communities during, or just after, periods of major stress. For example, it seems likely that Jews became truly monotheistic only after their return from the Babylonian exile. At the time, it is likely that the returnees numbered only a few thousand. It is only in the following centuries that the monotheist Jewish headcount grew through natural population growth and conversion. That growth itself was indeed probably bolstered by religious factors. Newly minted monotheism is a very dynamic and expansive phenomenon. If we use the terminology of population dynamics, we might say that there is a very strong founder effect (a narrow population bottleneck) involved in the emergence of monotheism but that a population carrying such a cultural trait enjoys a strong competitive advantage thereafter. All this seems to be quite obviously applicable in the case of Judaism, Christianity and Islam but also probably in that of Zorostrianism and even of protestantism (seen as a monotheistic re-foundation of Christianity out of a a "polytheistic" catholic / orthodox substrate).
In the Roman context, all this points towards the early Rome rather than the late Empire. And indeed there is an early Roman event that looks suspiciously like a "monotheistic" reform : the suppression of Kingship and the establishment of the Republic. The earliest accounts we have of this episode were written at the end of the republican period, more than 5 centuries after the fact, by members of the senatorial class who had grown accustomed to think primarily in political rather than religious terms mostly under Greek influence. As a result, the "establishment of the Republic" is presented chiefly as being politically (and morally) motivated. However, there are some elements in the traditional account that betray the presence of strong religious forces at work. For example, we are told that the temple of Jupiter Capitolinus (the chief temple in Rome until Constantine) was just being built in its first incarnation precisely at that time. Before that, the Capitoline hill was covered by a myriad of shrines to local spirits and deities. Upon the construction of the Jupiter temple, all these shrines were destroyed and not simply to make space. Not only those lying on the top of the hill were removed, which would have been sufficient, but also those located on the slopes. Also we know through archaeology that a complex of temples (the Sant'Omobono area) built at the base of the Capitoline hill was abandoned (and maybe destroyed) at about the same time. Finally, it seems that in its early republican form, the senatorial class was much more a religious institution than a political one. Even in the late republic and the Empire, the members of this class had a hereditary monopoly on most of the high priestly functions of the Roman religion.
In other words, it seems that early republican Rome was quite similar to post-exilic Jerusalem. In both cases "polytheistic" shrines were being removed while a big temple was being built, on the top of a hill, in the honor of a single, all powerful male God (Yahweh / Jupiter). In both cases, this was being done under the direction of a hereditary priestly class (levites / senatorial class) governed by a council of elders (Sanhedrin / Senate) headed by a chief priest (Kohen Gadol / Flamen Dialis). The main differences seems to be that Roman early republican "monotheism" was not aniconic and, probably most importantly, did not develop a canon of scripture. Thus it was much more flexible than its Jewish counterpart as its (oral) history could be "re-written" at will to fit changing circumstances and new influences.
Starting from this, I believe it would be quite fun to try and develop a TL with a fully monotheistic Rome from the start of its (republican) history. What about, for example, a reversal of the roles of Rome and Jerusalem ? A (pagan) Jewish empire encompassing all of the Mediterranean basin and oppressing a tiny monotheistic Roman state ?
We could imagine Jewish religion "reverting" to paganism for some reason and have a Temple of Yaweh / Asherah / Astarte on the top of mount Zion (paralleling the temple of Jupiter / Juno / Minerva of mid-republican Rome) by 500 BC. Imagine that ! And the rest is history ...