The Eastern Orthodox interpretation of this passage is not that John personally asked this question because his faith was wavering; but rather that he sent his own followers to ask the question (11:2) so that they would begin believing in Jesus and follow Him.
The Gospel explanations by Theophylact, an 11th century Byzantine, are perhaps the best source of concise Eastern Orthodox interpretations of the Gospels. [Commentaries were in Greek and often point out subtleties in the Greek text that are missed in most Bible translations]. The King James translators sometimes consulted Theophylact when they encountered a difficult passage.] Theophylact was almost always summarizing much older commentaries from the Church Fathers, especially John Chrysostom. He comments on this passage:
Now when John had heard in the prison the works of Christ, he sent two of his disciples, and said unto him, Art Thou He that cometh, or
do we look for another?
John did not ask as if he himself did not know Christ. How could this
be when he had borne witness to Him, saying, "Behold the Lamb of God"?
[John 1:29]. But because his disciples were jealous of Christ, John
sent them to acquire more evidence, so that by seeing the miracles
they might believe that Christ is greater than John. This is why he
himself pretends to ask, "Art Thou He that cometh?" that is, He Whose
coming in the flesh is awaited in the Scriptures. Some believe that
by saying, "He that cometh", he was asking about the descent into
hades, as if not knowing the answer, John were questioning, "Art Thou
He that goeth even into hades, or should we look for another?" But
this is foolishness, for how could John, who was greater than the
prophets, not know of the crucifixion of Christ and the descent into
hades, when he had called Christ the Lamb Who would be sacrificed for
us? John knew, therefore, that the Lord would also go down into hades
in the soul so that even there, as St. Gregory the Theologian [Gregory
of Nazianzus, 329-390] says, He might save those who would have
believed if He had become incarnate in their day. John did not ask
this because he did not know the answer, but rather because he wanted
to provide his disciples with the evidence of Christ's miracles.
Look, then, how Christ answers the question ... [cf. Matthew 11:4-6]
The Explanation of the Holy Gospel According to St. Matthew
Two questions have arisen regarding this explanation:
Why did John the Baptist needed to "pretend" instead of simply telling his disciples, "Go and look at all the miracles He performs"?
Why did Jesus asked the disciples to “go back and report to John..." (Matthew 11:4) while He knew that it was all about the John's disciples rather than John himself?
The answer to the first question lies in understanding that John's disciples were intensely loyal to him and jealous of the attention that Christ was receiving at, in their eyes, John's expense. This is evident, John Chrysostom observes, in their earlier "complaints":
And they came to John, and said to him, “Rabbi, he who was with you beyond the Jordan, to whom you bore witness, here he is, baptizing,
and all are going to him.” (John 3:26)
Then the disciples of John came to him, saying, “Why do we and the Pharisees fast, but your disciples do not fast?” (Matthew 9:14)
Jesus Himself testifies of their obstinance when, knowing their lingering doubts, he concludes his talk to them saying And blessed is he, whosoever shall not be offended in Me (Matthew 11:6).
John Chrysostom's explanation for the Baptist's feigned curiosity is that his disciples obstinance concerned him, since he would not be around much longer to guide them toward Christ. Thus, he needed to do something that would require them to speak with Christ in person. He writes:
For as yet they knew not who Christ was, but imagining Jesus to be a
mere man, but John greater than after the manner of man, were vexed at
seeing the former held in estimation, but the latter, as he had said,
now ceasing. And this hindered them from coming unto Him, their
jealousy quite blocking up the access. Now so long as John was with
them, he was exhorting them continually and instructing them, and not
even so did he persuade them; but when he was now on the point of
dying, he uses the more diligence: fearing as he did lest he might
leave a foundation for bad doctrine, and they continue broken off from
Christ. For as he was diligent even at first to bring to Christ all
that pertained to himself; so on his failing to persuade them, now
towards his end he does but exert the more zeal.
Homily XXXVI on the Gospel According to St. Matthew
The second question is why did Jesus tell John's disciples to go back to him, since He would have known John's purpose in sending them. The answer, I think, is that they were still not entirely convinced and needed additional reinforcement from their own teacher. If the case were otherwise, Jesus would not have added And blessed is he, whosoever shall not be offended in Me. Theophylact paraphrases Christ's words, "'Tell John what you see, and certainly he will use the opportunity to bear witness more fully to you concerning Me.'" John Chrysostom also further explains:
For if He had said, "I am He," both this would have offended them, as
I have already said; and they would have thought, even if they had not
spoken, much as the Jews said to Him, You bear record of Yourself
[John 5:31; 8:13].
Since answering this question, it was modified to ask for interpretations of this Scripture by "the three main branches of Christianity".
I would submit that the "Orthodox" interpretation is essentially the (or at least "an") interpretation of the first millennium Church Fathers (at least that's what the Orthodox Church strives for).
John Chrysostom is a pre-schism Church Father and continues to be recognized as a Doctor of the Roman Catholic Church. I assume that later Roman Catholic interpretations will be in concert with his.
Most prominent Protestant commentators, to my knowledge, also respect Chrysostom's opinions. John MacArthur, for example, calls him "perhaps the greatest preacher of the early church".
Ellicott, however, allows that John the Baptist might have himself wavered, as "week after week passed without the appearance of the kingdom as he expected it to appear." Matthew Henry expresses hope, but no firm opinion: "We hope that John's faith did not fail." The Cambridge Bible for Schools and Colleges also enumerates three possible reason's for John's inquiry - (a) despondency; (b) to confirm the faith of his disciples (Chrysostom's interpretation); or (c) to force Jesus to declare Himself the Messiah - but it does not indicate which of the three is the best interpretation.