Amputations of limbs following battlefield injuries has been performed for centuries. It was thought better to have a relatively small wound than a wound that compromised a large amount of the limb. A skilled surgeon could remove a limb in around ten seconds, and after that, stitching up and bandaging the stump would take a minute or so.
Yes, there are risks. Asepsis wasn't recognised until the eighteenth century, but amputation was still a better risk than leaving an extensively wounded limb attached.
Now, removing wings may depend upon how they are attached. If they are attached to the ribcage by an independent scapula, it may be faster - and by extension, preferable - to remove them complete with the scapula, as the scapula is effectively floating, and can be removed in under a minute - I can attest to this personally, as in a University Zoology practical class, were asked to dissect a sheep, and were challenged to remove both a foreleg and a hind leg, and were timed while doing so. It took me well under a minute to remove the foreleg and scapula with a not-particularly-sharp 3" pocket knife.
On the other hand, if the wings are attached more solidly, perhaps to the same scapula as the arms, or to an intermediate pelvis-like structure, with a ball-and-socket joint, then either the ball joint would have to be disarticulated, or the bone cut. Cutting a bone can be done in a matter of five to ten seconds with the right tools, but with only the aforementioned 3" folding pocket knife, it took me around a minute to disarticulate the aforementioned sheep's femoral joint and remove the hind leg, and was much more of a hack-job than removing the foreleg.
Considering that the subjects of these amputations will not be anesthetised, the other point to amputating rapidly was to minimise pain. Pain may also be managed by exposing the subject to loud noises at the critical moment, as a matter of distraction.
Alcohol is a good disinfectant when distilled, but it is a very poor anaesthetic, as the dose required to achieve anaesthesia is very close to the lethal dose, and given the lag between administering it and its taking effect, it is very easy to kill the subject.
Opiates and cannabis can provide good pain relief even if they can't provide complete anaesthesia - if they are available.
So, if your winged people want to engage in this barbaric practice, the best way to minimise deaths and trauma is to use the services of a person with much experience and skill in performing the procedure - that way, the time in unanaesthetised surgery will be minimised. Asepsis would be good, but they likely won't think of it, and a lot of lives will be lost unnecessarily, even though many more will be saved.
However, to save the greatest number of innocent lives, the winged folk descending from the sky to run a spear through the bodies of the clergy and their assistants who would perform such barbarism, declaring as they do so that God does not approve - play up the Guardian/Avenging Angel aspect - would be the most effective course of action, though this is not exactly what the OP asked.