Rav Ovadia Yosef zt”l paskened that they are Jewish according to the Radbaz who said that they are Jewish but Karaites. If we go according to Rav Ovadia Yosef who based his teshuva on the Radbaz then they are Karaites. In fact, the Beta Israel practice what in Israel is refered to as “Haymanot” which is not Rabbincal Judaism which in that case would be the religion of the Karaites.
The Ethiopian Beta Israel who converted to Christianity (most, bit not all, were forced conversions) in the 19th and 20th centuries and their descendants are called the Falash Mura. At this time the Falash are the majority of Ethiopians who came to Israel and they are still, for the most part, practicing Christians. Rav Ovadia Yosef also considered them Jews, but I wonder if the Radbaz would pasken the same as he did to the Beta Yisroel who did not practice Christianity at the time. In fact, the Falash Mura themselves did not consider themselves to be Jewish until after the Beta Israel’s first wave of immigration to Israel.