The church at the outset of the fourth century had integrated into the Roman society surrounding it. While still being illegal in the Empire, Christianity was in reality so much a part of the society that it was no surprise when it was legalized in 313 AD under the emperor Constantine. In fact, the real surprise had been the Emperor Diocletian’s "Great Persecution" from 303-305 when Christianity, still technically illegal, actually came under attack. The early church had seen the Empire as the pagan world and something from which to be separate, but the fourth-century church viewed itself as both Christian and part of the Empire without feeling a contradiction between the two, as “the leaders of the Christian Church…found that they could identify themselves with the culture, outlook and needs of the average well-to-do civilian”. This is evidenced by Diocletian’s logistical struggles in his persecutions: he “had to start by purging the army and civil service”, and finally despite his “well-organized and systematic effort, …[he] had to admit defeat”. When Diocletian’s successor Constantine legalized Christianity it was a logical, practical step that further united the Empire and helped him solidify his throne as well as keep his army and bureaucrats.
(quotes referenced: R. MacMullen, Christianizing the Roman Empire, p. 44. J.F. Kelly, The World of the Early Christians, p. 194. P. Brown, The World of Late Antiquity, p. 82.)
Source: my undergrad senior history thesis
tl;dr By the time Constantine became a Christian he was doing it to fit in better with the people over whom he was trying to establish authority. It may have helped smooth over some political niceties, but Christianity had--and would have continued to--spread regardless.
You might enjoy this hour-long lecture by Professor Peter Heather on Constantine's Conversion and the Christianisation of Europe. YMMV, but I found it quite gripping and fun to follow (I love history but I don't like a snoozefest lecture any more than the next person).
It's been quite a while since I watched it though. I'll try and give some of the major points, and if anyone goes and watches it they can correct me if I misremembered:
In the context of there not being any conclusive evidence either way, Prof. Heather offers a counter-narrative to the idea of Constantine converting for political gain to a religion he didn't understand very well. Prof. Heather suggest it being also possible that Constantine was always a Christian, but simply hid his convictions when it was more expedient to do so. Iirc he had some interesting comments in general about the conflict in the early church between those who favoured publicly standing up for their beliefs (and risk being punished for it), and those who chose to keep their heads down and try to stay out of trouble.
On the effects of Constantine's conversion, he has some very interesting arguments that the Imperial court, and basically the entire culture surrounding the Roman Imperial court, was heavily focussed on unity to the point of sycophancy. Iirc he basically sketches and image where once the Emperor chose to become Christian, it was pretty much unavoidable that the whole Imperial court would follow in a show of loyalty and unity.
I think he was in general somewhat critical of suggestions that Christianity was on some kind of indomitable rise to power. I think his point was mostly that Christianity had shown itself to be a tenacious, vibrant minority religion but that it had stabilised and wouldn't necessaries have continued to grow without imperial support. But I really don't remember this last point very well and suspect I'm confusing Prof. Heather's arguments with those of other historians I read at the same time back then. So if someone has time to watch the lecture please correct me.
A follow-up to the OP's question--was there any realistic chance that the return of pagan emperors could have halted the spread of Christianity? ie, if Julian the Apostate had lived longer and re-established a tradition of pagan emperors