The Davidic Covenant is curious because of the unconditional nature of the covenant. Almost nowhere in the Christian world does anyone hunt down the people currently holding the covenant. This is a shame since it helps unravel other mysteries of the Bible.
The covenant with David was unconditional. He would always have some descendent sitting on a throne over at least part of ancient Israel.
19The word of the LORD came to Jeremiah: 20"This is what the LORD says: `If you can break my covenant with the day and my covenant with the night, so that day and night no longer come at their appointed time, 21then my covenant with David my servant -- and my covenant with the Levites who are priests ministering before me -- can be broken and David will no longer have a descendant to reign on his throne. 22I will make the descendants of David my servant and the Levites who minister before me as countless as the stars of the sky and as measureless as the sand on the seashore.'"
23The word of the LORD came to Jeremiah: 24"Have you not noticed that these people are saying, `The LORD has rejected the two kingdoms he chose'? So they despise my people and no longer regard them as a nation. 25This is what the LORD says: `If I have not established my covenant with day and night and the fixed laws of heaven and earth, 26then I will reject the descendants of Jacob and David my servant and will not choose one of his sons to rule over the descendants of Abraham, Isaac and Jacob. For I will restore their fortunes and have compassion on them.'" (Jeremiah 33:19-26 NIV)
19¶And the word of the Lord came unto Jeremiah, saying, 20Thus saith the Lord; If ye can break my covenant of the day, and my covenant of the night, and that there should not be day and night in their season; 21Then may also my covenant be broken with David my servant, that he should not have a son to reign upon his throne; and with the Levites the priests, my ministers. 22As the host of heaven cannot be numbered, neither the sand of the sea measured: so will I multiply the seed of David my servant, and the Levites that minister unto me. 23Moreover the word of the Lord came to Jeremiah, saying, 24Considerest thou not what this people have spoken, saying, The two families which the Lord hath chosen, he hath even cast them off? thus they have despised my people, that they should be no more a nation before them. 25Thus saith the Lord; If my covenant be not with day and night, and if I have not appointed the ordinances of heaven and earth; 26Then will I cast away the seed of Jacob, and David my servant, so that I will not take any of his seed to be rulers over the seed of Abraham, Isaac, and Jacob: for I will cause their captivity to return, and have mercy on them. (Jeremiah 33:19-26 KJV)
The word of Yahvah came to Jeremiah saying,20Yahvah says, If they can break my ordinances of the day and my ordinances of the night so that there should not be day and night in their appointed time 21then my contract which I made with David my servant may also be broken, so that he should not have an heir to reign on his throne, and my contract with the priests and the Levites my ministers. 22As the host of the skies cannot be numbered nor the sand of the sea measured, so I will multiply the seed of David my servant and of the Levites who minister to me.
23The word of Yahvah came to Jeremiah saying, 24Have you not considered what this people are saying, The 2 families which Yahvah has chosen, he has rejected? Thus they have provoked my people to anger, that they should not be a nation before me any longer. 25Yahvah says, If my contract not be in effect with day and night, and if I have not appointed the ordinances of the skies and land; 26then I will reject the seed of Jacob and of David my servant, so that I will not take any of his seed to be rulers over the seed of Abraham, Isaac, and Jacob, for I will bring them back from their exile and will have mercy on them. (Jeremiah 33:19-26 BRB)
Jesus Not Yet in Fulfillment
When the angel appeared to Mary she was told that her son would sit on David's throne. Jesus eventually will do this. But... It hasn't happened yet. Nowhere in the New Testament does the crown of David get passed on to Jesus.
Strategy for Finding
Though this can probably be tracked forward from ancient times, it is easier to simply use a process of elimination. Which throne in the world today is likely David's throne?
Understand that nearly all of the royal thrones of Europe are inter-married so any answer from the European field means all thrones in Europe are David's throne.
Simple answer is the throne in the UK. This appears to be one of the key redemptive gifts of the British people. The current Queen rules over an empire that has stretched across about 1/2 of the world.