In psychology, stress can sometimes lead to what is called Regression.
Regression involves taking the position of a child in some problematic situation, rather than acting in a more adult way. This is usually in response to stressful situations, with greater levels of stress potentially leading to more overt regressive acts.
Regressive behavior can be simple and harmless, such as a person who is sucking a pen (as a Freudian regression to oral fixation), or may be more dysfunctional, such as crying or using petulant arguments..
Source: Changing Minds.
Regression is a Freudian concept.
Someone that is using immature or babyish language while in a state of Regression is certainly doing something that could be described as "nonsense" within the context of the situation in which they are supposed to be acting like an adult.
Another possible term, also somewhat of a term of art, is Glossolalia.
Glossolalia or speaking in tongues, according to linguists, is the fluid vocalizing of speech-like syllables that lack any readily comprehended meaning, in some cases as part of religious practice in which it is believed to be a divine language unknown to the speaker.