In his commentary Henry Alford gives this analysis (his emphasis):
The Kingdom of the world (i.e. over this world: ἡ βασιλεία abstract. In the received text, reading αἱ βασιλεῖαι, it is the kingdoms, concrete, of the world) is become (aor., but alluding to the result of the whole series of events past, and not to be expressed in English except by a perfect) our Lord’s and of His Christ (no supply such as “the Kingdom,” is required: nor is this the case even in the rec. text. The gen. in both cases is one merely of possession), and He (no emphasis on He, as we are almost sure to lay on it, perhaps from the accent unavoidable in the Hallelujah Chorus of Handel) shall reign to the ages of the ages (this announcement necessarily belongs to the time close on the millennial reign: and this is no more than we might expect from the declaration of the strong angel in ch. X. 7) 1
Charles Ellicott adds to Alford's observations stating the Greek does not support placing emphasis on the pronoun "He" in "reigns:"
...our familiarity with the “Hallelujah Chorus” tempted us to put an emphasis on the word He which is not sanctioned by the Greek; it is the reign of the Lord which is the prominent thought. 2
Perhaps the best way to achieve the proper understanding is to remove “He" which may be implied but is not present:
The kingdom (βασιλεία) of the world is become our Lord’s and of His Christ and shall reign (βασιλεύσει) to the ages of ages.
βασιλεία is "not to be confused with an actual kingdom but rather the right or authority to rule over a kingdom." 3 In other words, it is the kingship (βασιλεία) being celebrated by the voices and the implication is there is a coregency of "our Lord and His anointed (see below)."
Heavenly speaking (voices) after the seventh trumpet follows the pattern of the seals and bowls:
When the Lamb opened the seventh seal, there was silence in heaven for about half an hour. (8:1) [ESV]
Then the seventh angel blew his trumpet, and there were loud voices in heaven, saying… (11:15)
The seventh angel poured out his bowl into the air, and a loud voice came out of the temple, from the throne, saying, “It is done!” (16:7)
In terms of voices in heaven, there is a progression toward the single voice. Given the knowledge there are many voices which can speak, we can say these voices which spoke after the seventh trumpet were silent after the seventh seal (and again after the seventh bowl).
After the many voices make their proclamation additional speaking follows:
And the twenty-four elders who sit on their thrones before God fell on their faces and worshiped God, saying... (11:16-17)
This is the sequence of speaking after a seventh item:
Seventh Seal Many voices and Elders are silent One voice is silent
Seventh Trumpet Many voices say... One voice is silent
24 Elders say... One voice is silent
Seventh Bowl Many voices and Elders are silent One voice says "It is done!"
What is said should be considered in light of who is speaking and what they are doing, but the identity of the many voices is not given. Whether the 24 elders were included is not stated (seemingly they are not as speaking appears to always be attributed to a source). Regardless, before speaking, the 24 elders get off their thrones and are never again described as sitting on or having thrones.
This is particularly relevant in context: following the many voices celebrating the reign in verse 15, the 24-elders leave their thrones (θρόνους). And they remain off for the remainder of the book. Thus, after the seventh item, John has also described a progression involving thrones:
Seventh Seal All thrones are occupied
Seventh Trumpet Many Voices proclaim the reign
24 Elders leave their thrones
Seventh Bowl One Voice from the throne
When the 24 elders leave their thrones, they direct their message to a single entity:
...“We give thanks to you, Lord God Almighty,
who is and who was,
for you have taken your great power
and begun to reign. The nations raged,
but your wrath came,
and the time for the dead to be judged,
and for rewarding your servants, the prophets and saints,
and those who fear your name,
both small and great,
and for destroying the destroyers of the earth.” (11:17-18)
This is addressed to Jesus Christ: He is the one who is and was and has begun to reign. He is the one who judges all things. He is the one who came to destroy the works of the devil and all those destroyers of the earth.
Therefore, in terms of the reign (v. 15), "our Lord" is referring to Jesus The Christ (as everywhere else in the New Testament) and His "christ" or anointed, would be King David.
The many voices should be understood as acknowledging the fulfillment of the Davidic covenant with coregents:
The kingdom of the world is become our Lord’s [Jesus Christ] and of His anointed/christ [King David] and shall reign to the ages of ages.
Thus both the pattern and type of King David's final reign mirrors its beginning:
The anointed King Saul The Anointed Jesus
The anointed King David The anointed King David
The LXX rendering of Psalm 2 has the nearly identical phrase "the Lord and his anointed" and some versions include the superscription (absent in the MT) stating it is a Psalm to David:
A psalm to David. Why did nations neigh, and peoples meditate upon vain things? The kings of the earth stood by, and the rulers gathered together against the Lord and against his anointed. (Psalm 2:1-2 ABP)
ψαλμός τω Δαυίδ ινατί εφρύαξαν έθνη και λαοί εμελέτησαν κενά παρέστησαν οι βασιλείς της γης και οι άρχοντες συνήχθησαν επιτοαυτό κατά του κυρίου και κατά του χριστού αυτού
του κυρίου ημών και του χριστού αυτού (Revelation 11:15)
The only difference is the voices in Revelation call "the Lord", "the Lord of ours."
The ABP also has a textual variant from the NU and M text at verse 11:17:
λέγοντες ευχαριστούμέν σοι κύριε ο θεός ο παντοκράτωρ ο ων και ο ην και ο ερχόμενος ότι είληφας την δύναμίν σου την μεγάλην και εβασίλευσας
The phrase "ο ων και ο ην και ο ερχόμενος" is a clear connection to Jesus (cf. Revelation 1:4, 1:8, and 4:8) and some manuscripts were made to make the connection between the Lord God Almighty (v. 17) and Jesus clearer.
1. Henry Alford, The Greek Testament, Volume IV, Moody Press, 1863, p. 665
2. Charles J. Ellicott, Bible Commentary for English Readers