What about the religious development? How would the demographics look like and how would the different sects of islam develop? also what would the caliphs religious views and approaches to other sects be like?
Basically, there would be no major Sunni-Shia split.
Upon taking power, the messianic fervour of the revolution would begin die down, as the new regime seeks to gain support of the moderate Muslim masses. As occured with the Abbasids OTL.
Though, this disappointed the Shia supporters of the Abbasids, who wished to see a divinely appointed messianic Caliph who would usher in a just Islamic society upon the earth.
Mansur was the complete opposite of this ideal, styling himself as a secular monarch, like that of the Umayyads, not appealing to the messianic imam-caliph the Shia were looking for. Leading them to look to the Alids to bring about the ummah led by a divinely guided imam. Though Mansur's son alMahdi and later Ma'mun pushed more for the religious connotations of their rule to win support of the Alids. Eventually leading to the state enforced Mutazila school, which gave the caliph ability to interpret the Qur'an and sunnah independently.
But ITTL, Abdullah alKamil had been cultivating the messianic/Mahdi status of Muhammad Nafs azZakiyyah for decades. So they would have more messianic overtones than the Abbasids.
Perhaps a 2 faced situation could occur, with Muhammad Nafs azZakiyyah appearing moderate in public, but to his private supporters being more messianic - these would be his bodyguards and most loyal troops, since they would see him as their Mahdi and be more than willing to sacrifice their lives, sort of like the Qizilbash to Shah Ismail. Overall, giving them a far more loyal powerbase than the Abbasids who only had the Khurasani who were mostly under Abu Muslim.....
This messianic and divinely appointed claim as well as the ability to interpret Qur'an and sunnah independently, would result in a much more authoritarian and despotic Caliphate. With no limits whatsoever on Caliphal power, due to it being perceived as divine will. Objectors may be put down by the zealots, since objecting to their Mahdi would be a form of blasphemy.....
OTL Jafar as-Sadiq taught Abu Hanifa and Imam Malik. ITTL, he could be appointed as the grand judge, thereby unifying schools of jurisprudence even more.
Muhammad Nafs azZakiyyah would try to include all branches of the Hashimids. Hassanids, Husainids, Abbasids and even descendants of Jafar ibn abi Talib whose great grandson Abdullah ibn Muawiyah led a major Alid revolt in the 3rd fitnah in 747.
With the majority of all governors throughout the Caliphate being from amongst the Hashimids - making the Caliphate based on the unity of the Hashimids family. (Saffah did the same thing, but kept it only within the Abbasid family)
By doing this, Shia disappointed with the regime don't have another major Hashimids to turn to, since they are all integral parts of the Muhammad Nafs azZakiyyah's government.
They may find a stray Hashimid from a small cadet branch and push their ideology upon him. But this wouldn't gain much support from the Shia masses, let alone the rest of the Muslim world.
Overtime these small groups around obscure Alids would become more ghulat/extreme like the Alawites or Druze or Qizilbash etc. But as they become more extreme, they would gain more opposition from the government who would eventually have to confront them.
A much larger problem would be succession disputes. Causing not only a political rift, but also an ideological one as occured with the Fatimid Caliphate. Which could result in completely different sects forming like the Nizari-Musta'li schism. Which may begin to split apart theologically too and if left long enough to fester could potentially break the state apart.
But generally, this Alid Caliphate would be less Shia than the Fatimids, since Shi'ism is still in its formative stages, and much of Fatimid Shi'ism's beliefs haven't been invented yet, and since the Alids rule the Caliphate said beliefs are unlikely to ever emerge.
The Khawarij would reject the Caliphate of Muhammad Nafs Zakiyyah, since they consider Ali as Kaafir. With the Shia outlet gone, this would probably be the dominant sect for ideological dissenters of the regime. Though I still see them as quite small, since kharijism never gained much popularity.
The Khawarij of the Maghreb (Sufris and Ibadis) would've already been co-opted into the regime by Ibrahim ibn Abdullah. Leaving their only dominant position in isolated Oman and Sistan.
Due to the significant Umayyad remnant ITTL, I could see the Umayyads developing an AbdManaf-id ideology. Since AbdManaf is the Father of Banu Hashim, but he's also the grandfather of Umayyah progenitor of the Banu Umayyah. Thereby including both Hashimids and Umayyads. This ideology becoming dominant amongst the Syrians who were the 3rd military power within the state after Berbers and Khurasani.
A few decades after the revolution, when the Umayyad rule is becoming a distant memory to the Shia, perhaps pressuring the government to adopt a portion of the AbdManaf-id ideology so that Umayyads are more integrated into the ruling class....
If the state tries to push Shi'ism too much, imposing an inquisition like the mihna, then there would likely be mass protests and civil unrest, as occurred with OTL mihna. Potentially the staunchly anti Shia Syrians could cause an armed revolt. So i don't see them pushing it too hard.
But overall, I see Muhammad Nafs azZakiyyah being accepted by the majority of the Muslim world. With no major ideological objections. Unless succession disputes result in theological schisms....
Demographics:
After the conquest of the Gangetic, Hinduism would be the largest religion in the Caliphate. Followed by Christianity, then probably Buddhism
(Central Asia would Buddhist. While Sindh and Bengal were also Buddhists though with significant influence from Hinduism) then Zoroastrianism or Islam, then Judaism then small religions like Manichaeism etc.
Not including north India, the population is probably around 50 million. The conquest of North India probably doubles that to 100 million. Over 1/3 of the world.
Of the Muslims, the Arabs are the majority, but second are the Amazigh/Berbers.
Their islamification was a unique process, mainly done by Abu Muhajir Dinar in 670s and Musa ibn Nusayr in the early 700s due to his manpower shortages. By 711, Tariq ibn Ziyad's invasion force being almost solely Berber. And by 740
all the Berbers seem to have embraced Islam, at least nominally, despite very few of them having ever met an actual Arab, since the Arabs were centred on Qayrawan, with almost zero settlement in the central or western maghreb (Algeria and Morocco respectively).
This means they were barely Arabised, knowing very little Arabic. Since they would be the largest army in the Caliphate, and dominate the entire west (from Andalus all the way to Egypt and Syrian coast), this could lead to tensions between the Arabs and their Berber overlords.
By comparison, the Khurasani army was mainly made up of Arab settlers who lived in Khurasan. While there was a significant portion of Iranians, these would've been much more Arabised than the Berbers, since Arab settlement was much larger and more dispersed in Khurasan, and marriage/integration of Arabs with locals seems to have been more common.
Though overtime, as cities are developed in the western and central Maghreb as well as maritime trade and industry, a large number of Arabs would migrate from the middle east to the rich Maghreb and Iberia. Which OTL didn't happen, since the abbasids didn't control past Ifriqiyah. Leading to urban Arab centres, which would gradually Arabise the berbers.
Since something like the Banu Hilal invasions wouldn't happen, as the Maghreb is much stronger and unified, the rural and Saharan populations of the Maghreb would probably remain much more Berber than OTL.
In India, it is an Arab Islam which has conquered northern India. Not the Persianate Islam of OTL under Ghaznavids, Ghurids, Delhi Sultanate and Mughals, who used Arabic as a liturgical language and Persian as the high culture language.
The sheer wealth of India would mean that much of the formative Arabic cultural efflorescence would occur in the metropolises of India, on par with the Iraqi contribution. With Indian Cultures heavily influencing early arabic culture, much more so than Persian culture, which would be relegated to Transoxiana/Khurasan and the poor and arid Iranian plateau, being on the receiving end of Indian cultural influences, instead of the giving end. Which is completely different to OTL.
This would mean Indian sciences, particularly the Hindu numerals, mathematics and astronomy of Brahmagupta, would enter the Muslim world much faster. Similarly the brilliant linguistic works of Panini would make Arabic linguistics more developed, as Panini came up with many advanced concepts which other linguists didn't come up with until the 19th century.
And Indian philosophies would be given more weight than OTL. Though as the Greek works are translated Indian philosophical though will have to compete with that of the Greeks. Perhaps creating more interesting and dynamic philosophies.
Islamification of India would be slow and gradual. With Muslims only being dominant in the cities, as in most regions. Though conversion would probably faster among the already declining Buddhist populations of Sindh and Bihar/Bengal, whose urban trade networks were being usurped by the newly arrived Muslims.
If a major Arab Bedouin migration (similar on scale to Banu Hilal) to the Thar desert, then much of Rajasthan, Sindh, southern Punjab, northern Gujarat and other regions surrounding the Thar desert would arabise and islamify significantly faster. Fromwhich regions surrounding those would also Arabise and so on.
Bedouin typically had little Islamic knowledge, and followed tribal law. So in India, after mixing and integrating with the local Hindus, perhaps some aspects of Hinduism or Rajput tribal law will affect these Bedouin?
Much of India at this time was covered in this forest. So if a policy similar to that of the Mughals in East Bengal: clearing forest, establishing agricultural villages headed by a Muslim, underwhom the inhabitants of the village would islamify. Then much of India could be converted. But this would still take centuries.
Indian traders would probably dominate the city of Basra, and other Persian Gulf, Arabian Peninsula and perhaps red sea ports. So there may some Hinduism practiced in those cities privately.
At the time, India had a huge connection to South East Asia, which had significant Buddhist Hindu influence. So Hindus may migrate to SEA in significant quantities to escape Muslim rule. Or Muslim traders may begin the islamification process of SEA centuries earlier. Or if a direct invasion of Srivijaya occurs from Caliphal Bengal, similar to the Chola's, for more control over the trade routes.
Then the Caliphate itself would be present in SEA, appointing governors and administering the region. Islam now challenging Hinduism and Buddhism, both of which were still in fairly formative stages, as was the political landscape with the first major kingdoms beginning to emerge now and most of their fertile farmland still being beneath dense forest, gradually being cultivated, increasing wealth, population and cultural developments as a whole. Islam being introduced now in such a formative era could potentially prevent the rise of Hinduism and Buddhism.
SEA seafaring technologies like the incredibly fast Trimarans and Catamarans could brought westwards along with Southeast Asian sailors, for much quicker communications between Andalus and the middle east or India and Iraq. Strengthening central control over these distant regions.
The early Caliphate having outposts in Sumatra and other regions of SEA would allow much more trade with China. As well as with Korea and Japan which OTL didn't receive many Arab traders. Japan was in quite a formative stage, so the interactions with Muslims traders could cause some significant butterflies, culturally and religiously
As for the court culture, then this would be heavily influenced by that of the Tang after the arrival of Princess Qizhaoyi and the Cui clan. Since they had the most developed court and administrative structure on the world at the time.
Probably with Chinese classics and Confucian texts being translated, affecting Arab culture
As for Europe, then most of it is still pagan: the Basques, all the Slavics, the northern and some of the eastern Germanics, Avars, Picts, Finno-Ugrics etc.
ITTL the united Caliphate controls Iberia and Maghreb, making their position in Europe much stronger. And could potentially aid these Pagans against their Christian enemies/neighbours in exchange for conversion.
As for the Basques, the perhaps using them as a minority rulers over the romance speaking Christian majorities of upper Spain and southwestern France, if they convert. Basque naval expertise could also come in great use to the Caliphate for Atlantic raids.
Otherwise it would be slow and gradual. Though, being connected to the central Muslim world means more Arab/Muslim migration to iberia. Which history only seems to have received 2 influxes: in 711 initial invasion and 742 when Balj ibn Bishr escaped the Berber revolt. It wasn't centuries later when Berber migration took place in the late 900s, though solely as military forces. With larger migrations after the AlMoravids conquest.
Since it's connected, and the Maghreb is unified, safe and prosperous, more Muslims would migrate to Iberia, making its conversion faster.
Unlike OTL, in which post 13th century Sufi mystics mainly spread Islam to India and SEA, with a great deal of syncretism taking place until reform movements in the 19th century, early Islam was much less like that, so the Islam of these disparate regions would be more theologically similar to one another.
Thus, in a much more expanded and unified empire, the travels of hadeeth scholars in the 9th century would be much more extensive. Instead of being dominated by Persians (all 6 major hadeeth books are by Persians), other groups would take a much more important role. Especially the Berbers and northwest Indians. As well as the more distant Andalusis and the northeast Indians.
This greater interaction should unify Islam more on the whole. (It would also give some interesting Nisbas) As well as make it's judicial system more robust, since there are more people's living in varied lands and cultures which have different problems than what OTL peoples of the Islamic world.were going through. Problems/situations which would need to be addressed by Islamic jurisprudence.