TL;DR They weren't Judaized Berbers, but instead refugees from the Jewish-Roman wars and rebellions. Carthage was important for Jews during late antiquity, and throughout the middle ages they established autonomous or independent tribes in the highlands to escape persecution along the urban coast.

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While the whims of warlords can be at times fickle, the answer almost certainly lies in the presence of the Carthaginian Jewish community. It is worth noting now that there was not exactly an outpouring of Berber converts to Judaism, and conditions that lead to tribes converting to it are quite rare indeed. Other examples of the conversions of tribes to Judaism tend to be seeking a middle-ground between great powers, such as between the Byzantines and Persians, but for Africa we have no such competition so-to-speak.

Back to the Carthaginian Jews for a second - the Judeo-Roman historian Josephus writes that the city of Carthage owed its foundation to Jews. While this is unlikely given that Carthage spoke Phoenician, built Phoenician, and claimed a Phoenician heritage for itself, it's not entirely out of the question that Jews had a long-standing presence in the culturally-similar city. Oral tradition, though not considered the most reliable per se, has the Tunisian Jewish community arriving in the area following the destruction of Solomon's Temple and fleeing from the Babylonians. This might explain Josephus's claim a little bit, but it's hardly solid evidence of anything really.

Speaking of solid evidence, the first properly solid evidence we have of Jews in Carthage dates to the Roman Imperial era. While the Josippon would suggest that Titus moved Jews to Carthage after his pillage of Jerusalem, our grounded evidence begins a bit later. Whatever the case, this was a very important era in Jewish history: this was the Talmudic era, where the oral traditions and the arguments of sages were being written, codified, and commentated, to preserve Pharisaic teachings following the destruction of the Second Temple.

The Talmud offers us the names of several sages from Carthage, and archaeological studies have unearthed a significant Jewish necropolis. The Jewish presence in the city was non-negligible to say the least. The material culture left behind appears to be quite romanized, which would be a trend for the city at this point in history. The significance of this community should not be understated, and a part of them would move to Djerba and establish a community there as well. The Djerba community and its synagogue, though reconstructed relatively recently, is a source of pilgrimage for its great antiquity and the amazing continuity of its congregation.

These Carthaginian Jews would be the first step for establishing the Berber Jews, but are hardly the only step. This early establishing of Jews in North Africa gave a foothold, in essence, that would allow for further migrations in the near future. As for your question as it pertains to antiquity, Judaism in the African interior would emanate from the Carthaginian community.

The reason I'm being tentative with my answer and putting so much trust in Carthage is thus: I've not heard of there being too many examples of Berber Jews before the rise of Islam. During this period, whatever limited Jewish presence was there, it was certainly funneled out from the coastal urban centers. The rise of Islam across the region and subsequent politics and economics through the early middle ages brought the greater bulk of Jewish presence and history to the region. The Islamic expansion brought with it Jewish merchants and craftsmen, and the Visigothic conquests in Spain also drove Jewish refugees into Islamic Africa where they were better received at that point in time. As there formed distinctly Jewish tribes in Yemen, so too did the Atlas Mountains see the rise of distinctly Jewish tribes, not made of converts, but instead of Jewish refugees from Carthage, the Levant, and Southern Europe, come together and forming their own communities in the rural parts of the land. Judeo-Berber came to be the dominant spoken language, and Jews lived among Berbers relatively well for quite a long time. These communities would continue to be bolstered as groups like the Almohads would drive Jews out of the cities, and even further as the Spanish Inquisition brought even more refugees to Africa.

The consistency of Jews as the original ethnic stock for these tribes is agreed upon with Ibn Khaldoun, a medieval Berber scholar from modern Tunisia. Ibn Khaldoun writes clearly that the preislamic Berbers were by and large not Judaized, that the Jewish tribes he encountered were, in fact, Jews and not Bebers, and that the few Berber tribes who might've become Judaized subsequently adopted Christianity in late antiquity. This is supported by modern genetic evidence.

In short: the Jewish Berber tribes were none other than Jews themselves, adopting the Berber language(s) and living in the African highlands where they could be relatively safe and autonomous. This is not a unique phenomenon among the Jews of the Islamic world, and such can be seen also in Yemen, where distinctly Jewish tribes lived autonomously and gained reputations as silversmiths, engravers, and mercenaries without equal. They even became royal bodyguards! Starting with the early migrations out of Carthage as either a long-established community, or recent refugees from the Levant, they began a steady flight to the rural Atlas highlands to live autonomously and were bolstered or even defined by later migrations during the Islamic era.

Great answer, thanks!