It can be hard to generalize about traditional/school grammar — different books and different teachers have different idiosyncratic views and terminology — but I don't think that most traditional grammars would recognize either of those nominals as a constituent worth mentioning. (That may sound strange coming from a modern linguistic perspective — how can you just, not mention a constituent? — but traditional grammars just aren't that precise about most things.)
I'll cover a few examples below, to give you a flavor.
If you're asking this question because you want to be able to refer to this constituent in a traditional grammar context, then I would suggest using "noun phrase" both where CGEL uses "noun phrase" and where CGEL uses "nominal". Traditional grammar generally uses "X phrase" to mean "a multiword phrase that functions roughly like a single X", so if "the old man" is a noun phrase and "old man" is a constituent, then "old man" is a fortiori a noun phrase as well.
Now, for the promised examples . . .
Robert Lowth's 1762 Short Introduction to English Grammar, page 106, treats examples like "the old man" as follows:
8th Phrase: When the quality of the Substantive is expressed by adding an Adjective to it: as, "a wise man;" "a black horse." Participles have the nature of Adjectives; as, "a learned man;" "a loving father."
Note the idiosyncratic term "8th Phrase" (so called because he puts it eighth in a list of different types of "Phrases" — a very haphazard list by modern standards), and the absolute non-mention of articles. I'm actually not 100% sure if the substantives here are supposed to be "man", "horse", and "father", or "a man", "a horse", and "a father"; if Lowth were around to ask, I'm guessing that he'd be surprised at the question, since the articles aren't the important part of the examples.
A century later, Alexander Bain's 1863 English Grammar, page 163, gives the example of "The neglect to lay down in distinct terms the opposition between true and false, has been […]"; it says that the subject is "neglect", and that it has two "attributive adjuncts", namely "the" and "to lay down in distinct terms the opposition between true and false". It gives no indication of whether Bain regards "neglect to lay down […]" as a constituent and "the neglect" as a non-constituent, and if he were around to ask, I'm not sure he'd have an easy time of it.
A century and a half later, Grammarly's online article "Everything You Need to Know About Sentence Diagramming, With Examples" says that articles are "also a kind of modifier", and says to "Place modifiers and articles on diagonal lines beneath the words they describe." You can see in example #3 that in the phrase "his old ball", "his" and "old" are rendered the same way: they play the same role, namely, they both modify "ball". I will make no guesses about the individual author's awareness of modern linguistics, because he didn't invent this diagramming scheme (it matches what I recall from English textbooks in the 1990s) and I don't assume that he endorses it; but suffice it to say that nothing in the diagramming scheme, or in the text of the article, suggests the notion that "old ball" might be a constituent within "his old ball".
NP) from old man (an
N′, pronounced "N-bar"). As McCawley puts it, the N′ is part of the X-bar system, but the NP is not -- it's the grammatical equivalent of a logical argument, just as
Sis a grammatical equivalent of a logical proposition.