You can actually nasal breathe while swimming, though it’s not essential. Being in the water, your swimming environment is naturally humidified so the drying effects of mouth breathing is not so much of a concern. Also, if you are anything more than a recreational swimmer, the slight changes you need to make to your technique to accommodate proper nasal breathing (slightly lengthening your stroke to allow you to keep your head out of the water for a slightly longer period of time) is probably not worth the trouble.
However, if you do want to give it a go, it’s certainly possible. The key is to condition yourself to doing it slowly over time. I recommend just standing in waist deep water first, and having your head submerged in the water as if you were swimming, while practicing turning your head to inhale through your nose [I.e. getting used to the water in and around your nose etc.]
Then, once you’re ready to do it while swimming, don’t try doing it on every breath. If you breathe on every second arm stroke, breathe through your mouth for the first three breaths, and then on the fourth breath go for a nasal breath (you will find you might have to lengthen your stroke slightly to allow your head to stay out of the water a fraction longer). Continue swimming while nasal breathing just on every fourth breath. Once this becomes entirely comfortable, drop down to every third breath, then every second breath etc, until you can nasal breathe on each breath.
Once you can do this get ready to get some really strange looks from lifeguards and people watching you from the side of the pool. It’s hilarious! They think you are not breathing, or some sort of freak show.
But Wait – !!!
While I encourage you to try nasal breathing during swimming, it’s certainly not essential. As the main purpose of the exercise, according to Ayurveda, is to get into the exercise high or mind-body zone state, you don’t want your breathing to be strained in any way. This can happen with nasal breathing during swimming. I swam while nasal breathing for about two years, but have since given it up, as I found it took my attention away from the effortless, easy (zone-out) flow of swimming. I still do it a bit in warm-ups and cool downs, and definitely recommend it here. However, if you find it a bit laboured during your entire swimming session, simply focus on taking mouth breaths deep into your belly, and breathing out through your nose during your exhalations.
Hope that helps, and happy swimming.