Part of the issue with this is the fall of the Ilkhanate can be mainly chalked up to the various epidemics that ravaged Iran in the early-mid 1300s which decreased their legitimacy, leading to the rise of the Jalayarids, the rebellion by the various Anatolian Beyliks and the return of Chagatayid power in areas like Afghanistan and Kashmir (leading eventually to the Timurid Empire). I do not see a way to save the Ilkhanate without avoiding the Black Death in Iran or Iraq at least.
Either way, the Ilkhan is an amazing example of a state built effectively and on firm ground with great potential, that ultimately failed, its failure was sheer bad luck at an enormous scale.
With the Ilkhans continuing in power we see a less Persian dominated bureaucracy as seen in the Safavid period, yet more diverse (far) than the subsequent Timurid period and the continued use of Yassa as the legal ruling code of the land, as it was in say the Golden Horde Russia. Ghazan's reforms on currency and fiscal policy was extremely important and likely would continue had they not fallen, with by the mid 1370s, Iran being far better transitioned and economically viable than under the terror of early Timurid rule. The cities of Baghdad, Tehran, Istafhan, Basra, Delhi, etc would be far more populated, Iran in general would by the 1350s recovered from the invasions of Hulagu and Temujin, not be further depressed by the more vicious Timurid invasion. All in all both Iraq and Iran are far more prosperous in the 1400s and then into the 1500s and onward, and by extension of butterflying the Timurids as without the power base in Afghanistan and Peshawar, would not have the power to defeat the Tughluq Shah Nasir-ud-Din Mahmud at Delhi, depopulating the city and stressing the surrounding countryside.
In terms of religion, we do not likely see the Shi'i rise to particular prominence in Iran as they did with the Safavid and subsequent rulers. Shi'i in this regard will likely be the majority in the north around Gilan and the province of Mazandran and the in areas like Tehran and Qom. Sunni will dominate the south around Shiraz, Istafhan, Hormuz, Qeshm, Gwadar, etc... The populace of non Muslims is further far larger, with Christian tribalists and various forms of traditional religious groups living on the periphery close to Afghanistan, Buddhists remain a portion of the court as advisors on medicine and such. Christian slaves from Georgia and Circassian also are frequent in this tl. Possibly what we see in future times is Iran looking similar to Iraq unless a strong secular national identity is built, with sectarian violence being commonplace without a strong government entity and the absence of unifying secular goals.
The Ottomans likely still rise to power, conquering Constantinople earlier unless the Mamluks or Ilkhan takes interest. Though the Ottomans if they do rise, perhaps become even more so Greek, perhaps abandoning their Turkish heritage, as the Ilkhan will rule vassals to the direct east of the Ottomans lands and Ankara. In which case the possibilities for what to do with the Ottomans are endless.
Cilicia Armenia survives as long as the Ilkhan supports them. They continue raiding and attacking the Mamluks, making little headway. Armenians in general are more populous in Syria and Anatolia.
The Mamluks after their victory at Marj al-Saffar, had quelled the Ilkhan menace, though this isn't guaranteed to last. If the Mamluks do falter, they lose Syria and little else, as the area south of Dimshaq will be well guarded. However, more likely is continued Mamluk success and a far better Egypt, which recovers from the Black Plague and doesn't fall to the Ottomans. Portuguese trade wars are likely fought by an uneasy alliance of Ilkhanate, Mamluks and Gujarat as opposed to the Mamluk-Ottoman alliance of the period. Mamluks also in the future could foreseeably continue its cultural golden age, especially after a recovery in the 1500s.
Maybe this will stir a discussion.