summary:

It seems nasal breathing has multiple benefits and is not difficult to start (a few minutes per day for a few weeks). The easiest and most useful exercise: pause breathing repeatedly for 30-40 seconds. There are these objectives: stop mouth breathing, increase the time the breath can be held, decrease the amount of air inhaled. Among positive consequences: less pollutants in the lungs, better control of breathing and less tension, less oxidation, better O2-CO2 circulation, positive impact on asthma, sleep apnea, heart etc.

exaggerated claims: the techniques does not seem to “stop allergies”. The technique may improve breathing during allergic reactions, but probably of little help in serious cases. There are other claims that are oversimplified.

However:

asthma: numerous controlled studies show that the technique offers substantial improvement in a few weeks with relatively little effort.

sleep apnea: condition seems to be greatly improved by the technique.

tension: there are numerous reports that the technique reduces tension.

respiratory infection (common cold): nasal breathing shorten the duration and reduces symptoms (see 10. Breathing through the Nose (a simple exercise to clear up a stuffy nose)).

hyperventilation: there is some disagreement if the hyperventilation is a symptom of asthma, heart conditions etc. or a cause of them (it seems it is one of the causes). In any case there is a universal agreement that the hyperventilation is not healthy. The breathing technique clearly helps to reduce it. See normalbreathing.com, including this list, for some fairly persuasive scientific claims (mixed with some claims which are not based on research), and many published references.