It is too bad that the literal rending of the phrase "kingdom of heaven" is not used.
In Matthew, the first of the Gospels, as set forth in his Interlinear Greek-English New Testament, Dr. George Berry correctly translates it as "the kingdom of the heavens."
Daniel put it this way in Daniel 7:27, "Then the sovereignity, power and greatness of the kingdoms undr the whole heaven will be handed over to the saints, the people of the Most High. His kingdom will be an everlasting kingdom."
It is used only in Matthew, being specially in harmony with the purpose of the Gospel, to set forth Jesus as the coming Messiah to rule over the kingdom of (or from) heaven on the earth.
Certainly this Messianic Kingdom to come, is also in the sphere of the kingdom, or dominion, of God, which I believe includes the whole universe. And thus, "the kingdom of God" is used exclusively in the other three gospels.
Indeed, we in the Church/body of Christ are in the Kingdom of God. Our hope is in the heavenlies as revealed by Paul, especially in Ephesians chapts. 1 thru 3.
Bick