I believe that born-again Christians are a new creation and that God's kingdom has already come. Yet I pray, "May your kingdom come." The kingdom is past, present, and future. It is both now and not yet.
As far as the new heaven and new earth are concerned, I consider them to be the future aspect of the kingdom.
Yes. The view you presented falls under the purview of the "not-yet" concept held by Partial-Preterists, Futurists, Historicists and some Idealists. But under Fulfilled Eschatology, it is seen as a compromise. Full Preterists, Preterist-Idealists, and Realized-Preterism see themselves as strict adherents to Sola Scriptura.
Here's what they see...
"Thy Kingdom Come..." was still in the future at the time Jesus made the prayer, but is now fulfilled.
Jesus, during His ministry, said...
"But if it is by the Spirit of God that I cast out demons, THEN THE KINGDOM OF GOD HAS COME UPON YOU." —Matthew 12:28
Of course, the Jews were expecting something tangible, something they could see in order to believe, but Jesus upset their expectations by letting them know that the "Kingdom does not come with observation " (Luke 17:20-21). We will notice that Jesus does not follow it by saying, "but eventually it will be seen and will be observed", thus nullifying the not-yet expectation.
And so, in Fulfilled Eschatology, the "will" of God was done, and the Kingdom is a full present reality that's already here.
Likewise, the New Heavens and New Earth (all things made new) came at the regeneration of the Church, from a historical point of view. But we can also look at it from the perspective of each person that converts and comes to Christ. When someone becomes "born-again" or is "in Christ" then everything becomes "New"...
"Therefore, if anyone is in Christ, he is a new creation; old things have passed away; behold, all things have become new." —2 Corinthians 5:17
When once they could not see, or comprehend the Kingdom of God, upon being Born-Again they could...
Jesus answered and said to him, “Most assuredly, I say to you, unless one is born again, he cannot see the kingdom of God.” —John 3:3
Only until we've entered the Kingdom can we truly appreciate its reality...
Jesus answered, “Most assuredly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God." —John 3:5
It takes being Born-Again or being born in the Spirit to realize the Spiritual Kingdom of God...
“That which is born of the flesh is flesh, and that which is born of the Spirit is spirit. Do not marvel that I said to you, ‘You must be born again.’ —John 3:6-7