As discussed, Nessie doesn't come to the surface to breathe often. So it's likely she's developed gills - as I can't believe an animal has evolved that can hold it's breath for days.
I did a bit of research into shark's gills, since I was looking into sharks anyway. I've never fully understood how gills work, to be honest. The fish swallows water and then gets oxygen from it?
But actually, it's a remarkably similar process to breathing air. The water goes in the mouth and OUT through the gills - and the tissue in the gills acts like alveolar sacs in the lungs - absorbing oxygen into the bloodstream and passing it into the heart.
Plus, sharks have gill rakers, which stop food getting trapped in the gills, a bit like an epiglotus stops your food going down your air-hole.
Nessie, as a large, warm-blooded, aquatic creature, would struggle to get enough oxygen into her bloodstream with gills alone. Like a seal, dolphin or whale, she needs a lot of oxygen to keep her going, so I think that's why we see her at the surface sometimes. For huge gulps of air. But most of the time, she moves slowly, lazily in the cold, murky waters, and her extensive and complex gills do a perfectly fine job.