JOEL - I suppose God forgives us as soon as we have fulfilled all the steps of the repentance process and have proven to Him that we have forsaken the sin. The forgiveness is immediate; it's the repentance process that might take some time, which of course depends on the nature of the sin. Like you said after we have "really repented".
We must first recognize and confess that we have sinned ( Mosiah 26: 29, D&C 64: 7, 1 Jn. 1: 9); we must feel godly sorrow for the sin (2 Cor. 7:10), and make right what we did wrong. And we must ask God for His forgiveness and promise that we will not do it again (Mosiah 4: 2, 10). After all this and some time passes we will realize that we have overcome the sin and feel a change in our hearts and peace in our souls as the spirit witnesses to us that we have been forgiven (Mosiah 5:2, Isa. 1:18).

When Jesus healed the man with palsy He told the man, "thy sins are forgiven thee" after which the man immediately recovered and arose and walked home (Luke 5:18-25). Forgiveness was immediate in this case as evidenced by how quickly the man was healed. Enos in the Book of Mormon received an immediate forgiveness for his sins when he prayed to God for an entire day (Enos 1:1-8).
Of course these two people had the luxury of having Christ speak directly to them, letting them know that they were forgiven, something that does not often happen to the rest of us. There really aren't any scriptures that specifically state at what moment we receive forgiveness, but I also believe our loving Heavenly Father does forgive us immediately if we are truly repentant.

The Book of Psalms has a verse that you refered to which says: "As far as the east is from the west, so far hath he removed our transgressions from us." (Ps. 103: 12). This is refering to the fact that when we repent He removes the sin far away from us and remembers it no more (Isa. 43: 25, Heb. 10: 17, D&C 58: 42).