The Byzantine Empire fell in the 8th century rather than the 15th century. Without a powerful Christian buffer state to oppose it, Islam quickly spread throughout Eastern Europe. Constantinople became both the capital of the Umayyads and a major pilgrimage site that rivaled even Mecca and Medina.
The Abbasids, feeling the Umayyads had grown too decadent and forgetful of the teachings of Islam (and angered over how much money Mecca and Medina were losing to Constantinople) would go on to overthrow the Umayyads as they did in our world.
The Frankish emperor Charlemagne converted to Islam, both due to having several Muslim tutors in his youth and seeing that Islam would most likely become the next major religion. Charlemagne went on to declare Jihad against the Umayyads and anyone he perceived to be an enemy of Islam. After helping establish Abbasid power in the Middle East he went on to conquer the Italian Peninsula.
The Abbasids, realizing they'd inadvertently created a monster, had Charlemagne poisoned. The noble families of Rome, lacking a clear leader, established something similar to the senate of their ancestors. This led to the establishment of an Islamic Roman Empire stretching from the Pillars of Hercules to the River Ganges. Several Romans unwilling to convert to Islam, including the Pope, and several Umayyads, fled to Ireland. The Umayyads converted to Christianity and became the new ruling class of the island nation.
The Mongol Empire managed to last considerably longer than in our world due to the greater emphasis on central government. Although Tengrism is still the dominate religion, Buddhism is winning more coverts by the day. Missionaries from Ireland have ensured that Christians are also a small but growing presence in the empire.
The Mongols were able to successfully conquer Japan. Several Japanese fled prior to the invasion in hopes of establish a new Japanese homeland free of Mongol rule. They arrived in Mexico and quickly conquered several native peoples and establish a new empire combining Japanese and native culture and traditions.
Northern Europe (the parts not under Roman rule) is still pagan. The no alcohol aspect of Islam is somewhat hard to sell. This has prompted many Imans within the empire to reinterpret the rule that "no Muslim shall drink alcohol" to be "no Muslim shall drink alcohol to excess" to encourage the conversion of those territories. This has created considerable controversy within the empire.
Russia was established by Viking princes as in our world, and has served as a (somewhat redundant) buffer state between the Romans and Mongols after many years of passing back and forth between the two empires.
Britain has been balkanized (the in-universe term is "anglicized") between Rome, Ireland, Scotland and the Danish. The Midlands is more of less neutral territory and is the site of much cultural blending. Scotland is a mixture of Christians and pagans due to the work of Irish missionaries.
Several nations have experienced Industrial Revolutions including Rome, the Mongols, Mali and India. Srivijaya and Ireland are currently in the process of industrializing. Mali has remained strong and continues to expand. Though Rome initially wanted to conquer Mali they are now more than happy to have a fellow Islamic empire as an ally.
Srivijaya has become a major power in the South Pacific and Indian Ocean. Their empire has expanded and their trading fleets are found in ports in all corners of the world. The fact that they are primarily Hindu and Buddhist has caused some issues when trading with Rome and has lead to their alliance with Hindu India. They are also the primary trading partner (besides Japan-in-exile) with the Inca Empire.
Several powers are beginning to colonize the New World. The Vikings are expanding there colonies along the northeast coast of what would be North America. The Irish established a colony just in case Rome ever conquers them and to spread Christianity to the natives. They originally intended to settle around Massachusetts, but after discovering the Vikings were already there moved to Virginia and the Chesapeake Bay. Rome has also established a colony at the mouth of the Mississippi with its capital at New Alexandria (roughly where New Orleans in OTL).