The NT often refers to the two kingdoms in terms of the distinction between “this/the age,” (or the “present age”: which represents the kingdoms of the world) and the “age to come” (which represents the kingdom of God).
Most Christians naturally assume that the “age to come” begins when the “present age” ends. That is, one age ends and the other begins. The NT, however, contends that this is not the case.
The NT affirms that with the coming of Christ the kingdom of God/age to come has already arrived. This is evident in Mark 1:15, where Jesus announces that
“The time is fulfilled, and the kingdom of God is at hand.”
The presence of the kingdom of God is also evident in Jesus’ declaration: “But if I cast out demons by the Spirit of God, then the kingdom of God has come upon you” (Matt 12:28).
This means that there is a sense in which the “age to come” has already arrived. This is why Paul can claim, “They were written for our instruction, upon whom the ends of the ages have come” (1 Cor 10:11).
One significant point of confusion resides in the fact that, though we see that the NT affirms that the “age to come” has already arrived, the designation the “age to come” itself suggests something future.
Naturally, when the “age to come” was first used, it was a reference to a future time. The problem results from the fact that once the “age to come” arrived, the NT writers, understandably, continued to call it the “age to come.” (I suppose it beats calling it: “the age that formerly was called the ‘age to come,’ but shall henceforth be called ‘the age that has come and, yet, will come.’”)
Another potential point of confusion is that the “age to come” has arrived only in part. Perhaps the easiest way to understand this is to note that when the “age to come” arrives in totality, God’s presence will fill the whole of creation and death and suffering will cease to exist. Since death and suffering continue, we recognize that the “age to come” has only arrived in part.
Another difficulty arises in that, though the “age to come” (kingdom of God) has to some degree already arrived, “this age” (the “present age” or the kingdoms of the world) continues. That the “present age” continues is evident from the fact that sin, suffering, and death, all key features of “this age,” continue to plague our world.
This means, then, that in the present time both the kingdom of God and the kingdoms of the world exist.
 Matt 12:32; 13:39-40, 49; 28:20; Mark 10:30; Luke 16:8; 20:34; 1 Cor 2:6, 8; Gal 1:4; Eph 1:21; Tit 2:12 (note 1 Cor 1:20 uses “this age” but not in an eschatological context).
 Matt 12:32; Mark 10:30; Luke 18:30; 20:35; Eph 1:21; Heb 6:5.
 Paul does this to some extent in 1 Cor 10:11. The rest of the NT, however, continues the distinction between “this age” and the “age to come” likely because, as I will note below, the present time is a time in which both co-exist.
NB: Any subtle indication to Monty Python language is purely coincidental.
 This is a key claim of 1st John. Eternity has broken into the present: “What was from the beginning, what we have heard, what we have seen with our eyes, what we have looked at and touched with our hands, concerning the Word of Life—and the life was manifested, and we have seen and testify and proclaim to you the eternal life, which was with the Father and was manifested to us” (1 John 1:1-2). See: Donald Mills, ‘Eschatology of 1 John’ in Looking into the Future.
 The coming of the Spirit was a fundamental element of the promise of God and the key sign of the coming of the kingdom (“age to come”). See Ezek 36:27; 37:14; 39:29; Isa 44:3; note the presence of the Spirit was a key feature of the Messiah (Isa 61:1; John 1:32-33). The coming of the Spirit is a key sign throughout the NT that the kingdom of God has come.
 See Rev 21:3-4.