Hey! I wrote more in-depth about the Muslim conquests of Egypt in this post here but I'll copy the relevant portion to your question below.

Religion

This brings us to the second question of how was religion and language treated in the newly conquered region. First, it is important to know the state of religion within Egypt before the conquests. Although almost all of Egypt was nominally Christian, there were huge internal schisms between the Copts and the Melkites based upon the Council of Chalcedon, which ruled that Jesus was both 100% human and 100% divine at the same time. While the Melkites accepted this ruling, the Copts did not. Thus, each group considered the other as heretics, lacking the proper belief in a pivotal area.

This split is important as, post-conquest, the pro-Chalcedonian Melkites lost their basis of support in the Byzantium Empire. Thus, Coptic conversions to the Melkite church slowed substantially. Instead, conversions started to flow the other way, solidifying Coptic control on society. In the early years post-Conquest, these “Institutional Conversions” from one Christian sect to another were the vast majority of religious change in Egypt, instead of “Traditional Conversions” from Christianity to Islam.

Indeed, the Umayyad Caliphate did not, except very rarely, actively try and promote conversion to Islam. They even continued to discriminate against non-Arab converts by forcing them to continue to pay the jizya poll tax traditionally levied upon religious minorities. This discrimination against non-Arabs was a driving factor in the ʿAbbāsid rebellion in 750AD. Once in power, the ʿAbbāsids reversed this policy, exempting converts from the jizya and thus giving a strong financial incentive to convert. This lead to the first documented period of mass conversion from Christianity to Islam within Egypt. In total, the 9th and 10th centuries were a period of rapid conversions. By the end of the 10th century, Egypt was no longer a majority Christian nation. Additionally, these conversions were for the most part voluntary, with only a few high-profile cases of forced conversion being noted.

Part of the reason for the relative lack of conversions before the mass events of the 9th and 10th centuries was the societal pressures placed upon converts. Converts from Christianity to Islam (as with the few documented cases of conversion from Islam to Christianity) were often disowned by their families and forced to leave their hometowns. However, as the percentage of Muslims within Egypt grew (both from conversions and from Arab settlers), these societal pressures weakened, allowing more and more Christians to convert. This increase in conversions was then only strengthened with a decline in Christian education and instruction that blurred the worldview between Christianity and Islam.

So the conversion of the Christian Egyptians to Islam can be seen to be an overall gradual process, but one that substantially sped up by the 10th century, by which point Egypt was no longer a majority Christian nation (so during the Crusades, Egypt would have been majority Muslim, but not the 90-10 split we see today). While forced conversions occured occasionally they were not the primary driver of conversion, which was instead propelled by economic and social reasons. I don't know nearly as much about the Ottoman period, but hopefully another user can fill in the gaps. If you have any questions, let me know and I'll try to get back quickly!