The Idea in Brief
The third person masculine references in the second half of Dan 7:27 allude to "the Most High." That is, the Masoretic accentuation points to "the Most High" as the exclusive antecedent in the second half of the verse. For the same reasons, the Masoretic accentuation provides for the logical elimination of "the nation (of the saints)" from possible consideration.
Discussion
The Masoretes edited the Hebrew Bible from the eighth to the tenth century, and they placed short notes at the side, bottom, or end of the texts to provide clarification based on similar words found in the same or similar Biblical texts. Additionally, they also created a system of accentuation in the Biblical text. This system of accentuation not only helped to "sing" the text (and therefore aid in memorizing Scripture), but also helped to understand meanings. The British scholar Dr. William Wickes (1887) summarized the importance of the system of Masoretic accentuation as follows.
THE Hebrew accentuation is essentially a musical system. The accents are musical signs—originally designed to represent and preserve a particular mode of cantillation or musical declamation, which was in use for the public reading of the Old Testament text at the time of their introduction, and which had been handed down by tradition from much earlier times. . . . One marked peculiarity of the system could not, however, so long as the signs were accurately preserved, be lost. From the first, the aim had been so to arrange the musical declamation, as to give suitable expression to the meaning of the Sacred Text. For this purpose, the logical pauses of the verse were duly marked—and that according to their gradation—by pausal melodies, later by the accentual signs that represented those melodies; and where no logical pause occurred in a clause, the syntactical relation of the words to one another and to the whole clause was indicated by suitable melodies—partly pausal, partly conjunctive—and their corresponding signs. In this way, the originators of the system, and the accentuators who aimed at stereotyping their work, sought to draw out the sense and impress it on the minds of both reader and hearers. It need hardly be added that it is this, their interpunctional character, which constitutes for us the chief value of the accents.
In other words, Wickes indicates that the "interpunctional" character of accentuation is to provide the listener the logical "flow" of thought. In the Dan 7:27 passage, there are minor Masoretic notes in the margin, however, the system of accentuation provides more understanding as regards the logical "flow" of thought in this verse.
First, the Masoretic system of accentuation relied on the disjunctive accents to call out emphasis within the text, and therefore guide the logical "flow" of thought. The following list from Gesenius's Hebrew Grammar provides the principal disjunctive accents in descending order for the 21 prose books of the Hebrew Bible.

This hierarchy of disjunctive accents helps us to understand Dan 7:27. The more disjunctive the accent, the more acute the pause in logical thought. Thus we see that there are 10 disjunctive accents in this verse that help to guide the "flow" of thought.

As a side note, almost every verse in the Hebrew Bible is divided in half by the Atnach disjunctive accent. This accent provides the parallel of clauses (parallelismus membrorum or dialectus poetica), and thus helps to show how the Masoretes had understood this verse.
To begin, the first disjunctive accent (Geresh) occurs on the last syllable of the Hebrew word וְשָׁלְטָנָא. Please click here and please note the word וְשָׁלְטָנָא, which is colored in red. The Geresh signals to the listener that the words that follow are in logical connection, until another (more acute) disjunctive accent interrupts, which is the Revia.
Please click here and please note the word וּרְבוּתָא, which is colored in red. The Revia above this word signals to the listener that the words that follow are in logical connection, until another (more acute) disjunctive accent interrupts the "flow" of thought.
The next disjunctive accent, however, is not more acute -- please click here. This accent is the Yetiv, which occurs under the דִּי. This accent is not more acute than the Revia. If an accent appeared here that were more acute, then the "harmonic box" of the Revia (please click here) would be fractured. What ends the logical thought is the Zaqef Qatan.
Please click here and please note the Zaqef Qatan above the word שְׁמַיָּא, which is last word of the phrase colored in red. The Zaqef Qatan signals to the listener that the words that follow are in logical connection, until another (more acute) disjunctive accent interrupts the "flow" of thought.
Please note the cascading effect of logic as the listeners of public reading would "connect" the accents to the anticipated, subordinated lines of thought. This tracking enabled listeners to not only follow the words, but to follow how the ideas of those words connected within the verse.
The next disjunctive accent, however, is not more acute -- this accent is the Zaqef Gadol. Please click here and note the word which occurs on the last syllable of יְהִיבַת. This accent is not more acute than the Zaqef Qatan (especially since the Qatan precedes the Gadol). If an accent appeared here that were more acute, then the "harmonic box" of the Zaqef Qatan (please click here) would be fractured. What ends the logical thought is the Zaqef Qatan.
The next disjunctive accent, again, is not more acute -- this accent is the Tipcha. Please click here and note the word לְעַם is where this accent occurs. This accent is not more acute than the Zaqef Gadol. Also, please note, there is no logical connection with this word through any accentuation with the second half of the verse -- that is, this accented word is the "servant" to the last accent of the clause, which is Atnach.
Finally, the clause ends on the Atnach, which is the second strongest disjunctive accent. (The strongest disjunctive accent, the Silluq, is what is used to end verses.) Please click here and note the last word עֶלְיֹונִין in the phrase highlighted in red. (Note that the accent on the word קַדִּישֵׁי is conjunctive, and so "rolls in" to the עֶלְיֹונִין in the "flow" of logic.) The Atnach is the "foot-stomp" signal to the listener (and reader) that the remainder of the verse stands in logical contrast to the first half of the verse. Because this accent on "the Most High" (עֶלְיֹונִין) is stronger than the accent on "to the people" (לְעַם); and because the Munach accent on "the holy ones" is conjunctive and is the grammatical "servant" of the Atnach; the third person suffixes in the remainder of the verse "point" (through the system of accentuation) to "the Most High" (עֶלְיֹונִין) as their antecedent.
In this regard, the remainder of the verse appears as follows. The first three words are connected by conjunctive accents until the third word (עָלַם), which carries the Zaqef Qatan. Please click here. The Zaqef Qatan signals to the listener that the words that follow are in logical connection, until another (more acute) disjunctive accent interrupts the "flow" of thought.
The next disjunctive accent, however, is not more acute -- this accent is the Tipcha. Please click here and note the word which occurs on the last syllable of יִפְלְחוּן. This accent is not more acute than the Zaqef Qatan. If an accent appeared here that were more acute, then the "harmonic box" of the Zaqef Qatan (please click here) would be fractured.
What ends the logical thought (and the verse) is the Silluq - please click here. This is the strongest disjunctive accent, which is followed by what appears to be a colon (the Sof Posuq), and therefore terminates the verse logically and grammatically.
Conclusion
In summary, if we understand this verse through the Masoretic system of accentuation, the word "the Most High" (עֶלְיֹונִין) carries the Atnach, which is the strongest accent in the first half of the verse. The other words, "to the people" (לְעַם) and "holy ones" (קַדִּישֵׁי), on the other hand, carry lesser accentuation, and have no direct accentual relationship to the remainder of the verse. To illustrate, the following diagram provides the vertical perspective of how these accents interact with one another. The blue stars indicate the point in time when the hearers of the public reading of the passage would start to anticipate the complementary words and accents indicated by the red arcs. (The "harmony to the ear" would be grammatical, musical, and logical.) Please note that the word "the Most High" (עֶלְיֹונִין) has direct accentual relationship to the remainder of the verse.

In conclusion, the NASB therefore provides an accurate, and therefore very acceptable, translation of this verse in keeping with the Masoretic system of accentuation and cantillation.
Dan 7:27 (NASB)
27 Then the sovereignty, the dominion and the greatness of all the kingdoms under the whole heaven will be given to the people of the saints of the Highest One; His kingdom will be an everlasting kingdom, and all the dominions will serve and obey Him.’ [emphases added]
REFERENCE:
Wickes, William (1887). Two Treatises on the Accentuation of the Old Testament (Vol. II). Oxford: Clarendon Press, 1-2.