How did the Born Again teaching be associated with one first believing in Jesus Christ? Many Protestant Christians believe that you become born again as soon as you believe on Jesus Christ. Many believe that this teaching is clear in the opening verses of John chapter 3. How did this teaching get developed? Who started it? 

I'm not asking whether any church believes that a person becomes born again by believing in Jesus Christ. I want to know how the idea developed that born again was something that happened at one's first faith in Jesus, and not, for instance, at a later time in the spiritual journey. 

I'm acutely aware that I've accepted this teaching without question, probably as a result of group-think and not from personal theological study. Jesus doesn't plainly say that born again happens when you first believe on Him, so where did that teaching come from?  

An acceptable answer would be to say who developed the interpretation that Jesus' usage of born again takes place at the time when one first received Christ. Another acceptable answer would be to trace the development of the idea.
A common theme in the New Testament is rebirth in salvation through Christ. So the idea of being "born again" isn't all that cryptic. To specifically address your question of why we equate a new spiritual birth with the point of salvation, I'll direct you to a few scriptures.

First, we must clarify what the point of salvation actually is: 

Romans 10:9-10


  because, if you confess with your mouth that Jesus is Lord and believe
  in your heart that God raised him from the dead, you will be saved.
  For with the heart one believes and is justified, and with the mouth
  one confesses and is saved.


So if a confession of faith and a belief in the Christ is what brings salvation, then we can now compare that language to other scriptures that speak of being born again.

John 1:12-13 ESV


  But to all who did receive him, who believed in his name, he gave the
  right to become children of God, who were born, not of blood nor of
  the will of the flesh nor of the will of man, but of God.


Here we see a belief in the name of Jesus (point of salvation) being coupled with the idea of being born of God. We see this again in 1 John:

1 John 5:1 ESV


  Everyone who believes that Jesus is the Christ has been born of God,
  and everyone who loves the Father loves whoever has been born of him.


...and again in verses 4 and 5...

1 John 5:4-5 ESV


  For everyone who has been born of God overcomes the world. And this is
  the victory that has overcome the world—our faith. Who is it that
  overcomes the world except the one who believes that Jesus is the Son
  of God?


Again we see a tight coupling of two ideas: belief in Jesus and being "born" again into a spiritual family.

This same idea of "new life" is presented in several different ways throughout the New Testament. Paul tells us we are a new creation at the point of salvation:

2 Corinthians 5:17 ESV


  Therefore, if anyone is in Christ, he is a new creation. The old has
  passed away; behold, the new has come.


Again we see the idea of a "rebirth" -- since we are "in Christ" the moment we confess and believe, then it is at the moment of our salvation that we become a new spiritual creation. 

Something else we need to consider is that some scripture points to being "born again" as the starting point, with pursuit of holiness coming as a result of our initial rebirth. This seems to contradict the idea that being born again is something that comes later in one's "spiritual journey," as you put it. For example:

1 Peter 1:22-23 ESV


  Having purified your souls by your obedience to the truth for a
  sincere brotherly love, love one another earnestly from a pure heart,
  since you have been born again, not of perishable seed but of
  imperishable, through the living and abiding word of God;


Peter calls us to love one another earnestly as a result of being born again. It seems clear from this passage that the rebirth was the starting point, and Peter is referring to this past event as the reason for why we should want to pursue love in this way. 

As I mentioned earlier, a common theme in the New Testamanet is one of new life in Christ, and it seems clear that the idea of being born again fits with this new life we find in salvation through Christ. Sometimes this is referred to as being born of God or passing from death to life (John 5:24), but the message is quite clear that a life without Christ leads to death, and a life with Christ leads to a new life. Jesus even said it himself a few verses later in his conversation with Nicodemus:

John 3:14-16 ESV


  And as Moses lifted up the serpent in the wilderness, so must the Son
  of Man be lifted up, that whoever believes in him may have eternal
  life. For God so loved the world, that he gave his only Son, that
  whoever believes in him should not perish but have eternal life.


It seems unlikely that Jesus would so strongly emphasize the belief in the Christ as the initial point of salvation right after telling Nicodemus that somewhere along your "spiritual journey" you have to have this other event of being "born again" to see the kingdom of God. It doesn't make sense unless being born again is the actual point of salvation.