Peter has already responded, but I will weigh in too coming at it from a different direction. The origin of this 'stuff' claiming the British monarchy descend from King David is likely to have had one of two origins.
An Anglo-Saxon-superiority movement called British Israelism became quite popular in the early 19th century and would give rise to the white-supremacist movements in the United States (as well as some of its weirder religious cults). The central tenet of their belief system was that the Anglo-Saxons represented a Lost tribe of Israel, and were the only pure true inheritors of God's covenant with the Biblical Israelites, because the actual Jews were impostors and half-breeds. They came up with a whole chain of half-wit connect-the-dots links based on superficial word similarity and blatant misreading of the source material (of the sort of 'you can see the migration of the tribe of Dan in the names of the Danube River, Denmark and Doncaster'. Oh, and the Stone of Scone is really the stone pillow used by Jacob. Bat-crap crazy, nuttier than a fruitcake.
Well, to them, if they were actually Israelites then they needed a Davidic ruler, so they turned to the Book of Jermeiah where it reports that the daughters of king Zedekiah fled to Egypt, then to the "isles in the sea". That is obviously a reference to the British Isles (because what other islands are there, really?) so these daughters of Zedekiah must have been the ancestors of the British royal family because, . . . well, just because. It is all pretty silly.
The second possible origin for it comes from some sloppy scholarship of a few decades back, wherein the author concluded that the William of Gellone was actually a crypto-Jewish exilarch descended from king David. This got some play because William is likely an ancestor of the English kings (although I am not sure that any line is fully documented). Anyhow, the underlying thesis just represented wishful thinking, taking two contemporary people with different names and almost arbitrarily concluding they were the same. I have also seen another exilarch claim, going through a mistress of a Portuguese king that leads to the wife of George III. The critical link, as best I can tell, is completely made up, and this has fallen out of favor, replaced by an equally made-up idea that the mistress was really a black-African - don't ask, it would take longer to explain than it is worth, given that it total nonsense.
These are the only routes I have seen claimed, though there may be other claims.
taf