Alexander's Conquests, although swift, relied on a solid foundation in the League of Corinth. It depends on what Siddhartha can do. Unifying India seems like a difficult task unto itself— Siddhartha may have to play the Philip to Rahula's Alexander.
https://en.wikipedia.org/wiki/League_of_Corinth
However, the Shakya can easily extend their Ganasangha to a Confederation. Defeating enemy rulers, then admitting them into a government of equals with Siddhartha himself as a "Great Elect" (stole the title from his own ancestor, Maha-Sammata, who was also elected to rule the entire world, according to Buddhist cosmology).
https://en.wikipedia.org/wiki/Maha_Sammata
It would be incredibly funny if "Sangha" in TTL meant "United States" rather than "Monastic Order". In OTL, the Buddha modeled the Monastic Orders after the Senates of the Indian Republics, thus why they bore the same name.
Given India's syncretic and multicultural nature, I doubt anything as simple as Sun worship would stick around for long without innumerable layers of new religions slapped on top. That time period was rife with complex new schools.
https://en.wikipedia.org/wiki/Indian_philosophy
ADDENDUM: So what if the Shakya Empire became something like an Elected Monarchy? With only the descendants of Siddartha being eligible for "Great Elect"ship, and all of the defeated independent rulers are now "Elector-Princes" who have the right to vote for their overlord, but only from the pool of the appropriate bloodline? Seems considerably more stable than what happened with the Diadochi after Alexander's death.