I believe Wonderfizz may have been quoting Deut. 25:5 from the Luther-Bibel. Anyone interested can compare translations at, e.g.,http://www.diebibel.de/http://www.biblegateway.com/cgi-bin/biblehttp://www.hareidi.org/bible/bible.htm
The technical term for marrying a brother's widow is E levirate marriage, D Levirat, Leviratsheirat. It's an interesting sidelight, perhaps, but evidently a red herring re either Josefsehe or Onkelehe. (-:
So back to the original question: For the so-called Josefsehe (whose name seems to overlook both Jesus's siblings and modern historical scholarship on issues surrounding the so-called virgin birth, but <sigh> never mind), unconsummated marriage would be an understandable paraphrase. But I don't think there's a corresponding fixed expression in English, except maybe "marriage of convenience," or, more precise but less widely known, "mariage blanc."
Similarly, my impression is that Onkelehe (living with someone outside of marriage in order to continue to receive financial/death benefits from a spouse) must indeed exist (or have existed before legal reforms, as in Canada) in English-speaking countries. But to the best of my knowledge we don't have a name for it. For more background, try searching on terms such as widow(s), marriage penalty, survivor('s) benefits, etc.
In any case, I agree that the friend's relationship sounds more like a common-law marriage than either of those. (-: