Hey! I don't know where you heard that it took 1000 years for the majority of the population of Egypt to convert to Islam. In reality, the majority of Egypt was Muslim by around the end of the 10th century AD, around 300 - 350 years post-Islamic conquest. I wrote more about the conquest of Egypt here if you want to check it out and I'm going to paste the relevant section 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.
As for Iran, we can safely say that the conversion process was not so... gentle. There were two phases to Shia conversion in Iran. The first phase was rather short, lasting only from around 1501, when Shah Ismail took power, until 1514 and the Persian defeat to the Ottomans at the Battle of Chaldiran. During this period, Ismaili Shiism, or Sevener Shiism, was pushed by the Shah. Sevener Shiism is a minority sect of an already minority religious movement, being overshadowed by the much more common Twelver Shiism, but had several advantages for the Shah. Namely, he used its messianic leanings to declare himself as God's Messenger on earth. To enforce this, the Shah went so far as to punish disloyalty to the sufigare religious codes of conduct with expulsion from the order and, Ira Lapidus notes, possibly even execution. This harsh, even brutal, treatment was only a taste of what would come later.
After defeat at Chaldiran, Shah Ismail couldn't claim to be a messianic figure any more and so he officially declared the more mainstream Twelver Shiism as the state religion. First off, there were some structural changes. A state-controlled 'Ulema was established and Iran imported Shia judges and thinkers from the Levant, where Shiism had already been established for hundreds of years. The religious elite was then reinforced by large grants of untaxed, inheritable land. These measures together brought the most prominent religious scholars and local land-owners into the nascent Shia state's corner.
This was then coupled with campaigns of repression from various Shahs throughout the 16^th and 17^th centuries. Shah Ismail's own previous supporters who thought him to be messiah were killed, as were other extremists who were seen as a threat due to the possibility of their religious fervor turning them to rebellion. This same force was brought down upon the Sunnis of the kingdom - shrines were destroyed and the names of the first 3 Caliphs smeared - as well as the Sufis. The sufis themselves were driven out of the Kingdom in force by Muhammad Baqir al-Majlisi, the head of the Shia Ulema, at the end of the 17^th century, until only wandering ascetics remained.
The repression also extended other religions, such as Christianity and Judaism. Under Shah Abbas I, converts from these religions were able to seize the land and property of their non-converted family . Under Abbas II, wazirs were encouraged to force the conversion of Jews and, under Shah Husayn and the aforementioned Muhammad Baqir al-Majlisi, Zoroastrains were forcibly converted en masse.
I think it is pretty clear here the difference in how Egypt and Iran treated the conversion of their subjects. In Egypt, there were few cases of forced conversion ever recorded (I think I recall the number being something absurdly low like 4 total) and financial incentives that traditionally pushed People of the Book towards Islam, such as elimination of the jizya tax, were taken away. Compare this to Iran, where the State actively persecuted those who did not convert to Twelver Shiism (or converted to fervently to Sevener Shiism!), set up numerous financial incentives to convert away from other religions, and even forced conversions, notably against the Zoroastrians.
Sources
From Byzantine to Islamic Egypt: Religion, Identity and Politics after the Arab Conquest by Maged S. A. Mikhail
A History of Islamic Societies by Ira Lapidus.