I was reading about Mithraism on Wikipedia, and this sentence in the very first paragraph was really interesting.
"Mithraism, also known as the Mithraic mysteries, was a mystery religion centered on the god Mithras that was practiced in the Roman Empire from about the 1st to the 4th century CE. The religion was inspired by Iranian worship of the god Mithra, though the Greek Mithras was linked to a new and distinctive imagery, and the level of continuity between Persian and Greco-Roman practice is debated. The mysteries were popular among the Roman military."
https://en.wikipedia.org/wiki/Mithraism
It does not talk about how widespread was this cult and others. Also the article is very vague on how did this actually impacted the Roman army. I am not looking for a very in-depth reply, as this might be very hard question and very specific topic, so any pointers where I can read this up myself would be good enough.
Roman religion was less about faith and belief, and more about ritual and piety. It was very different from modern day religion. What you DO (rituals etc) was much more important than what you believed.
The Romans were big on rituals. Religion was less about what you believed, but more about rituals, and participating in public life. For instance, the imperial cult didn't really require you to believe in or subscribe to any belief, but you had to burn incense at their altar. At times, Christian refusal to do this was one of the main reason for the persecution.
This focus on ritual rather than belief allowed Romans to be members of multiple cults at the same time. You could be the devotee of the Greco-Roman God Bacchus, and at the same time be a member of the cult of Isis, and the imperial cult.
You didn't have to believe in any of these big cults, but participation in their rituals (especially the imperial cult) was more or less mandatory. It was seen as impious or even treasonous to refuse to participate
Religion was no different in the military. The Legionaries could be members of various cults. For most of the Empire's history, the cults of Mithras, Cybele and Isis seemed more popular.
Religion was particularly important in the military due to the Emperor (who was the commander in chief) being the Pointifex Maximus - the chief priest.
Religious rituals were often made before battle, where priests from various cults would make sacrifices, consult the auguries, or look for signs from the Gods, to bring favor to battle. The most famous of such a sacrifice was the "Devotion of Decius Mus", where he pledged his death in battle as a sacrifice to the Gods to bring victory to the Romans. Famous opposite cases include Publius Claudius Pulcher ignoring the bad sign of his chickens refusing to eat, and Crassus whose flag got "stuck in the sand", facing crushing defeat in battle after ignoring these bad omens
Gods were also used as propaganda, and Emperors would issue Gold coins with their faces and their patron god on it as an annual bonus to their troops, to show their devotion to their deity, and legitimize their divine support
I think changes in the role of cults started primarily with Aurelian, where he popularizes the cult of Sol Invictus. This was the first monotheistic religion to become popular in the Roman empire. From Aurelian onwards, the military had less respect for the individual deities for the smaller gods that individual soldiers believed in, and focused more on the emperor's main God (which at this point was just Sol Invictus, and later after Constantine, the Christian God)
TLDR; you are probably looking for the cults of Isis, Bacchus, Cybele, Mithras, and Sol Invictus. Gods were mainly used as propaganda via Gold coins, as the army's special bonus was paid with the Aureus. It was especially important after the Severan emperors ruined the silver coinage, leaving it as the only stable form of currency within the empire.
Follow-up: what would it take to be considered a cult, rather than just “normal” (maybe slightly more dedicated) worship of a specific god?