The Catholic Church held great power through most of its history due to its position as the mortal connection between man and God. The reckoning comes into play when you try to imagine a Catholic Church that could also demonstratively show its divine favor by healing wounds, curing sickness, and being able to perform multiple miracles a day. So let's reconcile this issue.
Look back to the hobby's roots in Chainmail and OD&D. A first level Fighting-man (or Fighter in AD&D parlance) was considered a veteran. This was a man who had fought in battles, faced his own death, and presumably killed a man. He was a trained warrior, not a neophyte facing battle for his first time. The precedent here is that even a first level character is a cut above the average person. Secondly, Clerics did not receive their first spell until second level. This was often justified as a requirement that a Cleric had to demonstrate his faith before his deity would deign to grant him divine power.
So let's extrapolate this a little. Most clergy in the EMP are not capable of channeling God's power into spells. Practically every manor had a chapel and every village a church of some sorts. The main spiritual requirement for this type of priest would be the ability to perform the sacraments (baptism and last rights at an absolute minimum) and lead prayer or conduct mass through some sort of missal or spiritual tract. Even after Gutenberg's printing press, Bibles were still not widespread and definitely still expensive. They were also still in Latin for the most part; vernacular was just beginning to enter into religion due to the efforts of reformers and 'heretics' in general.
A priest of this caliber is not going to be casting any spells. The majority of peasants and serfs would not be living in a community with a divine leader with the ability to perform miracles. You can see where this is going, the reconciliation here will be via scarcity. The potential to heal wounds miraculously and perform miracles will still be existent, but not common place. This feeling can be forced into a kind of anachronistic reality if you take the perception of a common man in that miracles really can be performed, but they usually manifest themselves as stories of how a man was saved over in another town by a traveling healer. We should only be so lucky, or faithful, or penitent, or flagellant, to experience one ourselves.
So who can cast divine spells? Mechanically, all Clerics can, as well as Druids and the other classes that gain spell casting at higher levels. In terms of lore, only someone particularly dedicated to God would have this ability:
- Traveling Friars. Friars were typically better educated and traveled than priests in a rural church. Historically, they preached directly to and lived among peasants, were invited by a local priest, and were required to give a portion of all offerings to the inviting Church. This could still lead to problems as a friar would be more liked than the local priest. For a fantasy setting, it makes for an excellent idea for a Player Character. While a PC won't be role-playing traveling and preaching, it does provide the context for why a Cleric would be adventuring.
- Monks And Monastic Orders: Monastic orders can represent people of particular faith and dedication so as to be able to cast spells. The long litany of orders, both Catholic and even some Reformed, provide plenty of room to create Divine casters that are still removed from every day concerns, preserving scarcity.
- Pardoners and Inquisitors: Pardoners offered indulgences to allow people to literally pay for their sins to be forgiven. They could have the ability to call upon miracles or could just be charlatans looking to fleece the flock. Inquisitions still occurred in the EMP, just none so infamous and widespread as the Spanish and Portuguese versions. In general, an inquisition was used to rout out heresy, and with the Reformation, there was plenty of heresy to be found. Inquisitors could have been granted spell casting ability due to their dedication to orthodoxy.
- Heretics And Reformers: Basically one in the same, depending on which side of the conflict you were on. A dedication to reforming religion, getting close to God, or a deep idealism could all be grounds for earning the right to cast spells.
That does leave one loose end, the raising of the dead. That will be covered in a post unto itself.