Yeah, Shiism is definitely present in pockets by that time, but are themsleves quite diverse. The most numerous groups would probably be the South Mesopotamians, with the most important political actor among them being the Musha'sha'iyyah of Khuzestan. The richest may be the Nizari Ismailis, who drew on trade/donations from others of their sect in India. The strongest militarily would be the wider Qizilbash movement-- but to keep their Safavid leaders from becoming temporal rulers, let's say the Aq Qoyunlu follow up on their defeat of Sheikh Haydar by occupying Gilan and getting his son Ismail I too. Any range of PODs are possible going all the way back to killing off
Haji Bektash in the 1200s before he can plant Shiism in east Turkey/Azerbaijan at all.
By the late 1400s the Aq Qoyunlu are the biggest blob on the map but their control is insecure at the center (succession crisis) and on the periphery (most of their Persian lands are governed in truth by former Qara Qoyunlu federates who joined Uzun Hasan to ward off Abu Said Mirza of the Timurids). They're not long for the world, which is why Ismail I was so successful in the first place. The Timurids aren't much better-- the last powerful member of that dynasty, Hossein Bayqara, concluded that Timur's realm could not be restored and focused on Khorasan while some petty lords (including Babur) feuded in Transoxiana.
No, absent the Safavids the likely rulers of Persia are the Shaybanid Uzbeks. Flung out of the imploding Golden Horde, they hung around the northern edge of the Timurid realm, interfering in the politics of Transoxiana and building up their strength. While Bayqara focused on Herat the Uzbeks swept south, mopping up Transoxiana and then Khorasan; Babur fled to Kabul. In OTL, Ismail's Safavids arose to oppose them, and after killing him (1510) sent his skull to Babur as a gift of goodwill. If they never arise, I'm sure the Shaybanids could keep riding right up to the Zagros before facing any serious threats. If Babur tries to challenge him from the other direction, Muhammad Shaybani has
his aunt and sister as hostages (and wives, but not at the same time).
A thing to note about the Shaybanids is that they are direct descendants of Genghis Khan, and this played a part in their propaganda about replacing the non-Genghisid Timur's line with "true" Khans. That sort of thing sold well in Transoxiana, don't know how it would go with the Persians (who have probably had quite enough of struggles for legitimacy among tribal foreigners). Muhammad's heir may paint himself as a more conventional Persian ruler (he would be half-Timurid, would be raised in Persia), but wouldn't revive the title of Shah-- he'd probably use Sultan to some limited extent and lean on Sunni rigorism as a source of legitimacy, in the mold of the Timurid Shah Rukh. That's assuming, of course, that this heir's succession isn't contested (a strong possibility given the fracturing of OTL Uzbek Transoxiana into 3 different khanates). The Sunni ulema remain influential in the intellectual and bureaucratic spheres; if later events in Bukhara are any indication, an ambitious scholar might even be able to take the throne at some point. Remnant Shi'i groups probably end up like Turkey's Alevis, but that's a better lot than Afghanistan's Hazara I guess.
In the best case, a Shaybanid Persia would retain land right north to the modern Kazakh border, and west to Herat (if not Kabul) as provinces, tied to the heartland by language, governance, trade, and religion. The natural bounds of Iran would be perceived as much larger, and let's be honest-- the Turkish influence wouldn't be that much greater than in OTL. In the long run the Uzbeks might even assimilate fully to Persian culture-- OTL they assimilated almost completely to the Karluk-speaking population of Transoxiana, leaving behind little but their name. It would be Timur's realm renovated, but court historians would probably harken back to Ghazan's Ilkhanate. Worst case, they endure an short and unhappy reign before being ejected by a rival west of the Zagros-- the Khuzestanis, another Caucasian powerhouse, or some other wildcard (a Kurdish principality?)