Does the Holy Spirit really come inside of a person at the time of conversion? Or, after true discipleship and the timing of God, does the Holy Spirit come in? However, during that time the Spirit is upon you?
It appears you are creating a distinction that is not supported by the Scriptures. Let me illustrate it by asking, when does Jesus really come inside a person? Does he do so at the time of conversion or at a later time of his choice? When does the Christian come inside Jesus, at the time of his conversion or at a later time of his choice?
I ask because Jesus stated, "At that day you will know that I am in My Father, and you in Me, and I in you" (John 14:20). Notice that Jesus describes being "in" as a two-way relationship. When does that relationship begin? "For as many of you as were baptized into Christ have put on Christ" (Galatians 3:27).
"For this reason I bow my knees to the Father of our Lord Jesus Christ, from whom the whole family in heaven and earth is named, that He would grant you, according to the riches of His glory, to be strengthened with might through His Spirit in the inner man, that Christ may dwell in your hearts through faith; that you, being rooted and grounded in love, may be able to comprehend with all the saints what is the width and length and depth and height -- to know the love of Christ which passes knowledge; that you may be filled with all the fullness of God" (Ephesians 3:14-19).
It is common to find people focusing on just the Holy Spirit, but notice that in this one passage we have the Holy Spirit, Jesus Christ, and the Father all in all the saints. People view being "in" as some form of possession, but what is being described is a close relationship of people living in close, intimate fellowship. See "Indwelling" for a detailed discussion of this topic. Suffice it for the moment to state that being "in" is no different between the Father being in the Christian, Jesus being in the Christian, the Spirit being in the Christian, or the Christian being in God.
But you ask specifically when this fellowship with the Spirit begins. "In Him you also trusted, after you heard the word of truth, the gospel of your salvation; in whom also, having believed, you were sealed with the Holy Spirit of promise, who is the guarantee of our inheritance until the redemption of the purchased possession, to the praise of His glory" (Ephesians 1:13-14). Peter stated, "Then Peter said to them, 'Repent, and let every one of you be baptized in the name of Jesus Christ for the remission of sins; and you shall receive the gift of the Holy Spirit'" (Acts 2:38). Paul stated, "For by one Spirit we were all baptized into one body--whether Jews or Greeks, whether slaves or free -- and have all been made to drink into one Spirit" (I Corinthians 12:13). Putting this all together, fellowship with the Spirit comes after hearing the Spirit's teachings, believing those teachings, and obeying His word -- specifically by being baptized as the Spirit taught in the New Testament. Yes, the timing is determined by God -- not individually, but by His plan for saving mankind from their sins.
Verses discussing the Spirit coming upon a person are fewer than you might think. It is used to mean that a measure of the Spirit's power is given to a person, such as Jesus' promise to his apostles: "But you shall receive power when the Holy Spirit has come upon you" (Acts 1:8). This was fulfilled not long afterward. "Then there appeared to them divided tongues, as of fire, and one sat upon each of them. And they were all filled with the Holy Spirit and began to speak with other tongues, as the Spirit gave them utterance" (Acts 2:3-4). Similarly, it was used to describe what happened to Cornelius' family and friends. "While Peter was still speaking these words, the Holy Spirit fell upon all those who heard the word. And those of the circumcision who believed were astonished, as many as came with Peter, because the gift of the Holy Spirit had been poured out on the Gentiles also. For they heard them speak with tongues and magnify God" (Acts 10:44-46). It is also something that happened in Ephesus. "And when Paul had laid hands on them, the Holy Spirit came upon them, and they spoke with tongues and prophesied" (Acts 19:6).
But the same phrase is also used when the Spirit is supporting a person. "If you are reproached for the name of Christ, blessed are you, for the Spirit of glory and of God rests upon you. On their part He is blasphemed, but on your part He is glorified" (I Peter 3:14). In this usage, it alludes to God's show of support for Jesus at his baptism. "When He had been baptized, Jesus came up immediately from the water; and behold, the heavens were opened to Him, and He saw the Spirit of God descending like a dove and alighting upon Him. And suddenly a voice came from heaven, saying, 'This is My beloved Son, in whom I am well pleased'" (Matthew 3:16-17).