While I in principle have no problem calling the phenomenon Judeo-Christianity as he suggests, I just don’t see what difference it makes. How does it provide any more clarity and sidestep the same pitfalls of the term “Jewish Christianity”. What’s more, the term “Judeo-Christian” is already used as a synonym for “Jewish Christianity” in current discussions—see the recent book entitled The Image of the Judaeo-Christians in Ancient Jewish and Christian Literature. I don’t think anything is gained by the term.
In addition, I have two further points in response to some of what Mike states. First we can all agree that the term "Jewish Christian/Christianity", and for that matter “Judeo-Christianity”, is a modern construct (not found in ancient literature) and is for many reasons an anachronistic designation for the reality of first-century Church (see Brant's comment below to my initial phrase "Jesus movement"). This is true not least for the decades of the middle to late first century when believers in Jesus were mostly of Jewish ethnicity on the one hand and Gentile God-Fears on the other who would have already been viewed as semi-Jewish. Therefore, we could define Jewish Christianity, and some have, as the Jewishness of early Christianity (see especially Daniélou's work).
Yet, since the time of Baur, Jewish Christianity has been variously defined and in the early days especially as the group who stood opposed to Paul's Gentile mission. This latter aspect for the most part has been jettisoned by recent scholarship, although some continue to propound it (e.g. David Sim). Nevertheless, what has essentially become the consensus is that the components of ethnicity and Jewish practice of certain kinds of believers in Jesus define Jewish Christian and not their perspective on Gentile Torah observance (see Mimouni). Of course on this question there was a wide spectrum of opinion.
Thus, in my view (and more on this in later posts) both Paul and his opponents (those who advocated Gentile conversion to Judaism), should be considered Jewish Christians—ethnically Jewish and practicing Torah (I am of the view that Paul didn’t cease his own halahakic observance in his mission to the Gentiles). On the law for Jews, Paul and the opponents seemed to have agreed. What distinguishes Paul’s type of Jewish Christianity, however, was not his view of the requirements for law observance on the part of Jews, but Gentile law observance. This issue was clearly a live question in the early church.
I prefer Skarsuane’s designation “Jewish believers in Jesus” to designate the ethnically Jewish believers in Jesus generally (including Jews who did and didn’t maintain a Jewish lifestyle) and “Jewish Christians” or Judeo-Christians as those who continue a Jewish lifestyle. See my discussion in an earlier post. Perhaps a further addition to Skarsaune's definition (and Mimouni's for that matter) may be to add that only were there differing Christologies among these groups, but also different perspectives on the status of Gentiles.
Second, I see no reason to wonder, as Mike does, who was more Jewish among the early Christian groups. What exact practices ethnic Jews who believed in Jesus preformed to my mind is not as important as observing who identified themselves (or were identified) as ethnically Jewish and were also recognized as such by particular Jewish communities at the time.