There are allegations throughout history that The Apostles distorted what Jesus actually did and said. This is an important issue because the claims formed completely new and separate religions. Mormonism and Islam both claim the Apostles screwed up and distorted what Jesus taught. Did they? What evidence does Islam and Mormonism have to support this claim? This is not about arguing the validity of one religion over another. This is solely focused on what we know about Jesus and what has been distorted about Jesus. Jesus is the primary subject and not any of the religions as a whole.
Islam & Jesus
Looking at Islam's claim it is easy to see issues. The quran claims that Jesus wasn't even sacrificed. Yet, by all other accounts, other religious (Jews) and non religious (Romans), he was. The person Isa' is about as historical reliable as Harry Potter. In fact, the only time Jesus is ever named Isa' is by Muhammad.
Then, according to Hadiths (Islamic Traditions), Jesus lived to be 120 and died naturally. The problem with the Hadiths is that they were written well after the quran (about 300 years after Muhammad) and even longer after Jesus walked the earth (900 years!) and the reliability is highly questionable as they contradict other known historical events, and misunderstand other religious teachings.
Some Hadiths and understandings of Jesus came from Gnostic sources like Gospel of the Infancy of Jesus Christ and The Gospel of Thomas the Israelite. Thus, they cannot be divine and given their distance from anyone even close to Jesus; they are not creditable or valid. Islam builds its case against the apostles with even less creditable sources. But even Islam has to account for why there is no grave of Jesus.
Islamic traditions claim Jesus received a book. Again, there is absolutely zero record of this except for what Islam claims. Muhammad failed to understand "the good news" message that Jesus taught and mistakenly thought it was a book. This book either never existed or was never found. Ironically the mistaken idea of Jesus having a book came from the synoptic gospels.
Muslims have absolutely zero witnesses who could validate any claims made by Muhammad or the Hadiths in regards to Jesus. In contrast, The Apostles could have been tested by their hearers and their hearers could have interviewed other eye witnesses and persons who had first hand knowledge of the teachings and events. Thus; the synoptic gospels are more logically and rationally reliable and trustworthy than the quran and hadiths in relation to the person of Jesus. Therefore the Apostles were the ones recording truth while Muhammad and later Islamic teachers distort what the Apostles recorded for them.
Mormonism & Jesus
In Mormonism, Jesus was created and did not exist eternally. That he was the product of relations between god and his goddess wife who used to be people from another world (McConkie, Bruce, Mormon Doctrine, p. 192, 321, 516, 589). Jesus is the literal spirit brother of the devil and of you and I (McConkie, p. 192, 589). This idea is a spin off from Arianism of the late 3rd century. But Arius taught a lot of things that even the Mormon Church would disagree with. However the Mormons claim this idea was revealed to Joseph Smith; yet Arius makes some of the same claims about the person of Jesus.
Mormons believe that The book of Mormon is more correct than the Bible, (History of the Church, 4:461). The true gospel was lost from the earth. Mormonism is its restoration, (Articles of Faith, by James Talmage, p. 182-185). "We believe the Bible to be the word of God as far as it is translated correctly . . . ," (8th Article of Faith of the Mormon Church). Their knowledge of Jesus, prior to Joseph Smith came from The Apostles. Even Joseph Smith's knowledge of Jesus initially came from The Apostles as well. Prior to Joseph Smiths "new" revelations, Jesus was already known by them. This proves that they first initially have to trust the synoptic gospels of which their whole spin off teaching comes from.
Much like Islam, there is zero support for their claims about Jesus as well... except from what Joseph Smith claims... kinda like Muhammad.
A mirror image of Mormonism, Jehovah's Witnesses also claim the Apostles distorted the truth about Jesus... and as did Muhammad... and so did Joseph Smith... and on and on other spin off religions can go. But if Mormonism, Islam, and Jehovah's Witnesses claim that Jesus existed and that The Bible is from God (in some parts and ways); then either God is unable to convey truth efficiently... or he did and He guided its record, because he is God.
Eye Witnesses & Jesus
John The Apostle, an eye witness of Jesus in the flesh, lived long enough to see the church begin to grow and spread. According to His disciples and their disciples, he wrote his eye witness statement. And we have a copy dating to 1 generation after his death. A copy that his disciples would have copied and given to their disciples. In it, he states (John 17:6-19):
6 I have revealed Your name
to the men You gave Me from the world.
They were Yours, You gave them to Me,
and they have kept Your word.
7 Now they know that all things
You have given to Me are from You,
8 because the words that You gave Me,
I have given them.
They have received them
and have known for certain
that I came from You.
They have believed that You sent Me.
9 I pray for them.
I am not praying for the world
but for those You have given Me,because they are Yours.
10 Everything I have is Yours,
and everything You have is Mine,
and I have been glorified in them.
11 I am no longer in the world,
but they are in the world,
and I am coming to You.
Holy Father,
protect them by Your name
that You have given Me,
so that they may be one as We are one.
12 While I was with them,
I was protecting them by Your name
that You have given Me.
I guarded them and not one of them is lost,
except the son of destruction,
so that the Scripture may be fulfilled.
13 Now I am coming to You,
and I speak these things in the world
so that they may have My joy completed in them.
14 I have given them Your word.
The world hated them
because they are not of the world,
as I am not of the world.
15 I am not praying
that You take them out of the world
but that You protect them from the evil one.
16 They are not of the world,
as I am not of the world.
17 Sanctify them by the truth;
Your word is truth.
18 As You sent Me into the world,
I also have sent them into the world.
19 I sanctify Myself for them,
so they also may be sanctified by the truth.
Either God ignored Jesus' prayer or all this is a lie; according to Muhammad, Joseph Smith, or Charles Taze Russell. For the synoptic gospels to be "corrupted" that would require the 'evil one' to get to their words and violate v15 which would also mean they were not adequately guarded (v12b) and God's protection failed (v12a).
30 Jesus performed many other signs in the presence of His disciples that are not written in this book. 31 But these are written so that you may believe Jesus is the Messiah, the Son of God, and by believing you may have life in His name.
24 This is the disciple who testifies to these things and who wrote them down. We know that his testimony is true.
25 And there are also many other things that Jesus did, which, if they were written one by one, I suppose not even the world itself could contain the books that would be written.
So, How Do We Know A Disciple of Jesus Wrote this?
Irenaeus of Lyons came to faith around 150AD and write his books around 180AD. He was born only 20 years after the Apostle John died. He was disciples by the direct disciples of John himself. In this writings he states:
“Matthew also issued a written Gospel among the Hebrews in their own dialect, while Peter and Paul were preaching at Rome, and laying the foundations of the Church. After their departure, Mark, the disciple and interpreter of Peter, did also hand down to us in writing what had been preached by Peter. Luke also, the companion of Paul, recorded in a book the Gospel preached by him. Afterwards, John, the disciple of the Lord, who also had leaned upon His breast, did himself publish a Gospel during his residence at Ephesus in Asia.”
Another awesome support is from Quadratus of Athens. Only a fragment of his writing remains. His work was written around 124AD. He was born before the turn of the century and died around 129AD. He is so close to the apostles that by some is even considered an apostle. He recorded:
"Our Saviour's works, moreover, were always present: for they were real, consisting of those who had been healed of their diseases, those who had been raised from the dead; who were not only seen whilst they were being healed and raised up, but were afterwards constantly present. Nor did they remain only during the sojourn of the Saviour on earth, but also a considerable time after His departure; and, indeed, some of them have survived even down to our own times."
Clement of Rome who lead the church in Rome around 88AD knew the Apostles first hand and in his writings quoted from theirs. Just like Quadratus, he would have known, seen, and able to interview and investigate eye witnesses of the events recorded in the Apostles writings.
Notice that all these (and there are more) recorded accounts are within 50-80 years of Jesus himself! The amount of resources to fact check the apostles claims was extremely accessible. Yet it is not until those who were healed, raised from the dead, and first hand witnessed the events had passed away that Gnosticism and spin off teachings began to raise. We have their records today! (besides having around the same time actual fragments and manuscripts of the writings themselves).
There is more recorded accounts from outside biblical sources that support orthodox Christianity and contradict any claim that the Apostles distorted the teachings and persons of Jesus.
The Jewish Talmud even records that Jesus was crucified in Sanhedrin 43 and that he had close disciples! And they hated this guy!
"On the eve of the Passover Yeshu (Jewish for Jesus) was hanged. For forty days before the execution took place, a herald went forth and cried, 'He is going forth to be stoned because he has practised sorcery and enticed Israel to apostacy. Any one who can say anything in his favour, let him come forward and plead on his behalf.' But since nothing was brought forward in his favour he was hanged on the eve of the Passover!35 — Ulla retorted: 'Do you suppose that he was one for whom a defence"
"Our rabbis taught Jesus the Nazarene had five disciples, and these are they: Matthai, Naqqai, Netzer, Buni, and Todah:
In other places of Jewish texts we see more mentions of Jesus:
Shabboth 14:4/8 – "someone ... whispered to him in the name of Jesus son of Pandera"
Abodah Zarah 17a – "One of the disciples of Jesus the Nazarene found me"
Sanhedrin 103a – "that you will not have a son or disciple ... like Jesus the Nazarene"
Sanhedrin 107b – "The master said: Jesus the Nazarene practiced magic
Even though they did not believe his teachings or even like him as a person; they openly admit Jesus did unexplainable things and called it 'sorcery' and magic, gained followers, had close Disciples, healed in his name, and then was "hanged". They validate orthodox Christianity without even knowing it recorded for all time.
Conclusion
Given all the outside sources and witnesses, unbelievers and believers, we can plainly see that later people such as Muhammad, Joseph Smith, and Charles Taze Russell have little to zero support for their teachings about Jesus. We also see the reliability of the writings of the Apostles as actual accounts of the person and works of Jesus. Thus we can know that the Apostles did not distort the person and teachings of Jesus and that we can even see for ourselves today.
In fact the only people distorting Jesus are the ones that invent later ideas about him and contradict those who were closest to Jesus.