I just had to deal with this exact response in a student yesterday, in fact ... it's rather common.
First off, understand why it's happening. We all have what's called a "mammalian reflex" ... which is a signal to the brain when the inside of your nostrils feel water to quit breathing. It's an instinctive reaction that's hard-wired into our brain. Some folks feel it more strongly than others ... but we all have that wiring. We are chronically nose-breathers after all, and have to train ourselves to breathe through our mouth while closing off the airway in our nose. This is just part of the training ... and like all training it comes harder for some than for others.
So how do we overcome it? Well, humans are delightfully adaptable ... and we adapt by doing. The most reliable method I've found so far is to have the student stand in chest-deep water, remove their mask entirely, close their eyes, and with regulator in mouth just bend over and breathe without a mask on their face. Take two breaths, then take your face out of the water. Wait a few seconds, now put your face back in the water and take three breaths. If the reaction doesn't overpower you, take more breaths. Continue this process, adding successively more breaths until you can stand there with your face in the water breathing off your regulator for a full minute. I guarantee you that each time it will feel easier ... you are "rewiring" your instinct.
Once you get to the point where it feels comfortable enough to breathe without a mask for a full minute, a final step is to drop down right where you are ... in water shallow enough to stand up if needed ... and do a mask removal and replace. This is good practice for more than just the mammalian reflex, since you're also learning how to deal with sealing your mask around your hood while wearing heavy gloves ... and even if you did this in the pool, doing it here in cold water is good training for compartmentalizing the effort to overcome the mammalian reflex by having to concentrate on the mechanics of dealing with your mask. Do this a few times ... until you can do it comfortably.
Now it's time to head out to the buoy or platform or wherever your class is doing their checkout skills and demonstrating that you have truly learned how to deal with this skill.
... Bob (Grateful Diver)