As I take it, "British Israel" theories are that the "ten northern tribes"of Israel populated Europe and the British Isles.  They generally leave out the "southern tribes" of Judah, Benjamin, and Levi.  That is how I have read their theories, although I have not spent a lot of time trying to validate them.  I have been interested of late in the "southern tribes" and their contribution to the gene pool.  My interest in this arose from a personal situation, in which my own genetic history became important to me, late in life.
I became interested because of the genetic quirks in my own physical "tent" (Paul the Apostle's word for our bodies).  It is of course fascinating to me that my mother's
mitochondrial DNA (and therefore mine and my son's) is J1C2b, and that J1C has a high likelihood of being descent from the "southern tribes".  So that means that Irish people like my mother, who was practically ghettoized Irish in genetic descent, have a large strain of Judah in them.  And as I said in my last post, this is only the maternal haplogroup; my mother's genealogy would have just as much of the "other" in it, presumably the northern strands which "married into" her haplogroup of thousands of years of J1c descent.

I am not even slightly interested in the concept that the northern Europeans are the Real Jews, as I think that is a half truth.  They have some Jewish descent, probably all of them do, which makes Hitler's claims ridiculous.  I am interested in this primarily as a genetic study, not as an assertion of spiritual hegemony.

And I think the Jews of Khazaria, who are often said to be "not real Jews, just converts" do have a spiritual claim to being Jewish, because for generations they have accepted the Abrahamic covenant, and conversion has ALWAYS been important in Judaism, with a few blips here and there when "blood" became the predominant qualification for being Jewish.  My husband's family has quite a history of dealing with "what is a Jew" and "conversion."  His daughter, who had some Jewish "blood",
had to convert to Judaism nevertheless in order to marry into her Jewish husband's family.  My husband, although half Jewish by descent, put his faith in Christ as predominant over that genetic background, and in fact was scarcely interested in his Jewish roots.  He was simply a Christian in his own eyes, which caused some anger and rejection in the family.  It is thus a personal thing to me, not an abstract study.  

As for the " British royal family", I do not think they have the Harp of David in their "coat of arms" for nothing.  Whether or not they think they have descent from one man called King David, they consider themselves spiritual descendants of that line.  And their insistence, up until recently, on marrying within the "line" shows that they were interested in their genetic descent.  The arrival of the gorgeous Catherine Middleton, with her non-line genes, is a magnificent breakthrough for them, but remember, they wouldn't consider any of her offspring royal if it were not for those pesky genes possessed by her husband.  They worship their genes, but maybe they can graduate from that.  

I am among those who believe that eventually this issue of who is a Jew and who is not will be completely evaporated by the Breath of the Spirit, and that in eternity if will
be as if it never were.