Daniel describes only the succession, not the time periods during which these kingdoms active or their duration. However, if we understand the second last Kingdom to be Rome, we must simply look to the Kingdom which conquered and overtook it.
What follows assumes the inspiration of the Gospel of Matthew, but that's not too much of a stretch for most here. But it becomes too obvious that Jesus Christ interprets Daniel's prophecy by His actions in Matthew 16 that it's not unhelpful to use this to interpret Daniel's intent himself.
In Matthew 16, we read several of Daniel 2's themes alluded to in striking detail.
Firstly, we have the King asking his servants something no one can possibly know by natural means:
Daniel 2:1-11 (DRB) 1 In the second year of the reign of Nabuchodonosor, Nabuchodonosor had a dream, and his spirit was terrified, and his dream went out of his mind. 2 Then the king commanded to call together the diviners and the wise men, and the magicians, and the Chaldeans: to declare to the king his dreams: so they came and stood before the king. 3 And the king said to them: I saw a dream: and being troubled in mind I know not what I saw. 4 And the Chaldeans answered the king in Syriac: O king, live for ever: tell to thy servants thy dream, and we will declare the interpretation thereof. 5 And the king answering said to the Chaldeans: The thing is gone out of my mind: unless you tell me the dream, and the meaning thereof, you shall be put to death, and your houses shall be confiscated. 6 But if you tell the dream, and the meaning of it, you shall receive of me rewards, and gifts, and great honour: therefore tell me the dream, and the interpretation thereof. 7 They answered again and said: Let the king tell his servants the dream, and we will declare the interpretation of it. 8 The king answered, and said: I know for certain that you seek to gain time, since you know that the thing is gone from me. 9 If therefore you tell me not the dream, there is one sentence concerning you, that you have also framed a lying interpretation, and full of deceit, to speak before me till the time pass away. Tell me therefore the dream, that I may know that you also give a true interpretation thereof. 10 Then the Chaldeains answered before the king, and said: There is no man upon earth, that can accomplish thy word, O king, neither doth any king, though great and mighty, ask such a thing of any diviner, or wise man, or Chaldean. 11 For the thing that thou askest, O king, is difficult; nor can any one be found that can shew it before the king, except the gods, whose conversation is not with men.
Matthew 16:13-15 (DRB) And Jesus came into the quarters of Cesarea Philippi: and he asked his disciples, saying: Whom do men say that the Son of man is? 14 But they said: Some John the Baptist, and other some Elias, and others Jeremias, or one of the prophets. 15 Jesus saith to them: But whom do you say that I am?
But which God in heaven alone reveals to men:
Daniel 2:27-28, 47 (DRB) And Daniel made answer before the king, and said: The secret that the king desireth to know, none of the wise men, or the philosophers, or the diviners, or the soothsayers can declare to the king. 28 But there is a God in heaven that revealeth mysteries, who hath shewn to thee, O king Nabuchodonosor, what is to come to pass in the latter times. Thy dream, and the visions of thy head upon thy bed, are these: .... ... And the king spoke to Daniel, and said: Verily your God is the God of gods, and Lord of kings, and a revealer of hidden things: seeing thou couldst discover this secret.
Matthew 16:16 (DRB) And Jesus answering, said to him: Blessed art thou, Simon Bar-Jona: because flesh and blood hath not revealed it to thee, but my Father who is in heaven.
A rock and the establishing of a new Kingom...
Daniel 2:34-35, 44 (DRB) Thus thou sawest, till a stone was cut out of a mountain without hands: and it struck the statue upon the feet thereof that were of iron and of clay, and broke them in pieces. 35 Then was the iron, the clay, the brass, the silver, and the gold broken to pieces together, and became like the chaff of a summer's thrashingfloor, and they were carried away by the wind: and there was no place found for them: but the stone that struck the statue, became a great mountain, and filled the whole earth. ... But in the days of those kingdoms the God of heaven will set up a kingdom that shall never be destroyed, and his kingdom shall not be delivered up to another people, and it shall break in pieces, and shall consume all these kingdoms, and itself shall stand for ever.
Matthew 16:18 (DRB) And I say to thee: That thou art Peter [Rock/Stone]; and upon this rock I will build my church, and the gates of hell shall not prevail against it.
All but explicit, Jesus even calls Peter the "stone" of Daniel 2!
The answerer of the question is elevated to a higher station and given authority in the King's Kingdom:
Daniel 2:6, 48-49 (DRB) But if you tell the dream, and the meaning of it, you shall receive of me rewards, and gifts, and great honour: therefore tell me the dream, and the interpretation thereof. .... Then the king advanced Daniel to a high station, and gave him many and great gifts: and he made him governor over all the provinces of Babylon, and chief of the magistrates over all the wise men of Babylon. 49 And Daniel requested of the king, and he appointed Sidrach, Misach, and Abdenago over the works of the province of Babylon: but Daniel himself was in the king's palace.
Matthew 16:19 (DRB) And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven.
Even more ridiculous is the fact that Peter famously founded the See of Rome, which Scripture calls "Babylon," as well as other ancient writings—see Revelation 17:5: "Babylon... [she] is the city that hath kingdom over the kings of the earth."
The kingdom begins small, but soon becomes the largest:
Daniel 2:35 (DRB) ...but the stone that struck the statue, became a great mountain, and filled the whole earth.
Matthew 13:31-32 (DRB) Another parable he proposed unto them, saying: The kingdom of heaven is like to a grain of mustard seed, which a man took and sowed in his field. 32 Which is the least indeed of all seeds; but when it is grown up, it is greater than all herbs, and becometh a tree, so that the birds of the air come, and dwell in the branches thereof.
Mark 16:15 (DRB) And he said to them: Go ye into the whole world, and preach the gospel to every creature.
Revelation 5:10 (DRB) And hast made us to our God a kingdom and priests, and we shall reign on the earth.
The Catholic Church alone fits this description in its particulars. It took a while, but it was foretold in the future tense: Rome was overtaken by the Catholic Church, the biggest claimant for the Kingdom of God, or the Church of Jesus Christ.