In Egypt, almost entirely. Already in the 10th and 11th centuries, Coptic, the previous vernacular language of Egypt before the conquests, was dying out. By the 12th "Coptic died out as a spoken language, and by the end of the twelfth century, Arabic had become the main written language of the Church. As is evident from the linguistic works of the great Coptic scholars of the thirteenth century, Copticwas already a classical language known only by those who studied it from preserved texts."

The same author then gives a very good overview of the situation elsewhere, which saw even faster Arabization:

Compared with the transition from Syriac and Greek to Arabic among the Christians of Mesopotamia, Syria and Palestine, the developments in Egypt are strange. While the Christians in these other areas started to translate their heritage into Arabic within a century after the Arab conquest and soon even to write theological treatises in Arabic, the Copts seem to have resisted any use of Arabic for almost two hundred years 6. But when the Copts gave in to Arabic, they did so much more thoroughly than any other Christians in the Middle East. Whereas Syriac, and to some extent Greek, at least as the spiritual language of the monasteries and the language of the liturgy, has continued to be widely used even till the present, Coptic died out almost completety.

Arabization was closely linked after the conquests with Islamization. Though hard data is difficult to come by, Richard Bulliett's estimates on conversion rates remain the most plausible that I've come across, which is to say that in most cases you're looking at conquered territories that were majority-Muslim within the space of a couple centuries after the conquests.

By the time of Mamluk rule in Egypt, the often Turkic-speaking Mamluks imported as slaves from abroad referred to their subjects, both Christian and Muslim, as awlad al-Arab, sons of the Arabs, as opposed to their own Mamluk backgrounds.

I'm far less familiar with the rest of North Africa unfortunately. Here, as with the Turks and the Persians Islamization did not necessarily involve Arabization as whole tribes might convert in the sparsely populated sahel, and yet the Berber-speaking population of North Africa remains very large. The same of course was true of these territories before the conquests.