I am not aware of literature that addresses the subject matter in the question directly. I have, however, thought about such cases quite a bit, and I have an unpublished paper that addresses some of the relevant data in part.
The interrogative word how is often a predependent of an adjective. It is asking for the relevant degree associated with the property expressed by the adjective, e.g. how big, how loud, how helpful, etc. These are relatively straightforward adjective phrases (APs), as pointed out in the question. I do not see anything mysterious or unusual about them.
Concerning examples with what, I think what often functions basically just like which, e.g. What apple do you have? = Which apple do you have?, What ideas is he promoting? = Which ideas is he promoting? In these examples, both what and which are functioning as interrogative determiners. On an NP analysis of noun phrases, they are predependents of the nouns that they introduce.
But the analysis of examples like what a beautiful child are indeed more challenging. They are challenging because it appears as though the noun child takes two determiners, what and a. This seems to be a particular idiosyncracy of what. Other interrogative determiners are incapable of doing the same, e.g. *Which a beautiful child.
However, English has a construction that seems related, e.g.
how big (of) a fool
too crazy (of) a man
that bold (of) a move
In these cases, the preposition of appears optionally. With what, the preposition of cannot appear, e.g. *what of a beautiful child. Furthermore, the indefinite article is obligatory in such cases. With what, the indefinite article need not appear, e.g.
what beautiful children
*how big (of) fools
*too crazy (of) men
*that bold (of) the move
These examples further illustrate that the ability of what to appear in NPs such as what a beautiful child is an idiosyncracy of what. One analysis of the construction might view what as a nominal that takes an indefinite NP as its complement: [what [a beautiful child]].
Finally, the following data can be construed as involving a type of ellipsis:
How crazy (she is)!
What a beautiful child (she is)!
What fun (that was)!
How entertaining (that is)!
The ellipsis mechanism optionally elides a definite pronoun and a form of the copula. This mechanism shows up in other constructions, e.g. If (you are) interested, you should contact me. I call this ellipsis mechanism slicing. I will share my unpublished manuscript on slicing if anyone is interested. I can be contacted via my email; my address is on my Wikipedia page: https://en.wikipedia.org/wiki/User:Tjo3ya.