Wasn't Luke a disciple of Paul and not Jesus though he did follow Jesus. So were the other three gospels written even later because Luke didn't come in the picture till 30-40 years after our Lord's death.
Luke is never described as being a disciple of Paul. Paul did not collect personal disciples. "For I have been informed concerning you, my brethren, by Chloe's people, that there are quarrels among you. Now I mean this, that each one of you is saying, "I am of Paul," and "I of Apollos," and "I of Cephas," and "I of Christ." Has Christ been divided? Paul was not crucified for you, was he? Or were you baptized in the name of Paul? I thank God that I baptized none of you except Crispus and Gaius, so that no one would say you were baptized in my name" (I Corinthians 1:11-15). Luke was a companion of Paul's who accompanied him on several of his trips.
The four gospels were written independently. When Luke wrote his gospel account would not reflect when the other three were written. Luke mentions there were other accounts. "In as much as many have undertaken to compile an account of the things accomplished among us, just as they were handed down to us by those who from the beginning were eyewitnesses and servants of the word" (Luke 1:1-2). Whether Luke is referring to the inspired accounts of Matthew and Mark or the personal writings of others is not known. Early Christian writers state that Matthew was written first, then Mark and Luke, and then John being last.
"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." [Irenaeus, Against Heresies, 3.1.1]
"Again, in the same books, Clement gives the tradition of the earliest presbyters, as to the order of the Gospels, in the following manner: The Gospels containing the genealogies, he says, were written first. The Gospel according to Mark had this occasion. As Peter had preached the Word publicly at Rome, and declared the Gospel by the Spirit, many who were present requested that Mark, who had followed him for a long time and remembered his sayings, should write them out. And having composed the Gospel he gave it to those who had requested it. When Peter learned of this, he neither directly forbade nor encouraged it. But, last of all, John, perceiving that the external facts had been made plain in the Gospel, being urged by his friends, and inspired by the Spirit, composed a spiritual Gospel" [Eusebius Pamphilius, Church History, 6.14.5-7]
The dating of Luke's account is based on the fact that Acts ends while Paul is still in his first imprisonment, about A.D. 62. The book of Luke was written before Acts, so a reasonable estimated date of writing would be about A.D. 60. The evidence in Acts, showing Luke traveling with Paul, is first shown in Acts 16:8-17 (the "we" in the passage). Acts 16 covers events around A.D. 50, so that would show that Luke was definitely a Christian before that time, which is less that 20 years after Jesus' death.
Matthew contains hints that the temple still stood when he wrote his account, so between this and the early Christian writer's statements an estimated date of somewhere in the A.D. 50's would be reasonable.
Peter died in A.D. 64, so Mark was probably written in the late 50's or very early 60's.
John also speaks of Jerusalem still standing (John 5:2 is in the present tense). John's account seems to be aware of the other accounts because he focuses on things the other counts did not mention or did not go into details about. Add to this that the early writers also mention that John was written last means that John was written some time in the A.D. 60's. The typical date is A.D. 65.