The history is a bit sketchy about this period. The most well-known early source, Ibn Abd al-Hakam Conquest of Egypt and North Africa and Spain, was written in 870. Being written about two centuries after the conquest, he had to rely a lot on oral traditions. In addition, many early sources focused on highlighting the Muslim victories and had little information on the conquered population and how they were treated.
That said, the Quran does prohibit forced conversions (Quran 2:256 explicitly says "There shall be no compulsion in religion"), and we do know that during the early Caliphates this prohibition was generally observed. That doesn't mean that the caliphates were fully tolerant by modern standard. Non-Muslim subjects were required to pay jizya, a special tax which also exempted them from military service. While jizya seems like a form of discrimination today, this served as an incentive for the rulers to keep the religious minorities thriving and not forcibly converted. This relative tolerance allowed indigenous non-Muslim minorities to exist today even in areas under millennia of Muslim rule, e.g. Egyptian Copts or Syriac Orthodox Church.
While Christians remained in a significant number in Egypt, in the Maghreb (Western North Africa) the number is much smaller, and even that might have been partially introduced by colonial powers. C. J. Speel's 1960 paper The Disappearance of Christianity from North Africa in the Wake of the Rise of Islam, argued that this was due to the fact that the Maghrebi Christians were Arians. They didn't believe that Jesus was God and thus were theologically closer to Islam than the mainstream Christianity of the Byzantine Empire.