Mewing is exercising the jaw muscles. What does that have to do with breathing through the nose?

No its breathing through your nose which forces your tongue to the roof of your mouth, this provides support to your nasal cavity and can change the entire shape of your face over time.

Mewing is essentially slang for correct tongue posture. Look up myofunctional therapy if you’re curious.

over how much time? The entire shape of your face changes regardless of what you do.

Timescale of years.

Correct tongue posture is a thing, much like sitting up straight it’s not going to change your life overnight but generally a good thing long term.

I don't believe sitting up straight is anything more than myth in terms of how intentionally doing it impacts your body

That is pseudo science, there is nothing to support it works. John Mew lost his dental license due to misconduct, the "mewing" movement should be seen like homeopathy movement.

And a few doctors ~150 years ago were ridiculed because they insisted that you should wash and disinfect your hands before surgery.

what? do people not breath through their nose?!

breathing through the mouth is hard and dries the lips / mouth