Davidic Covenant
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.
Background
The covenant with David was unconditional. He would always have some descendant sitting on a throne over at least part of ancient Israel.
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.