I also don't know much about the period, but I suspect that if the Mononobe manage to take control from the Soga, Japan would be less centralized and less Korea- and China-influenced. The Mononobe were much more suspicious of foreign things than the Soga, and believed Buddhism was anathema to the native Shinto religion; they would have fought Buddhism and certainly not promoted it like the Soga. Japan might be less Buddhist and centralized in future years and less willing to accept Chinese ideas and things like legal codes, Zen, and tea.
What I think I'm trying to say is the Mononobe might not just slow down the growth of Buddhism - they could stop it altogether.
If the Soga are completely trounced somehow, the main power struggle would likely be between the Mononobe and Nakatomi clans.
I'd really love to see an Asuka timeline someday. Do any exist?
I don't think on this board. I searched "asuka japan" and there were no timelines in the pre-1900 with a POD revolving around Japan. Do you want to collaborate on one then?
Okay, so how about this: Using a bit of Wikipedia research among other sources, it seems that the Mononobe were defeated by the Soga in a battle in 587. The point of divergence would be that the Mononobe in 587 manage to defeat the Soga instead, and wipe out the Soga clan. On the other hand, I've also encountered doubts whether there was this battle, or whether some event was dramatized.
Either way, this is a short while after Buddhism's reported 538 introduction to Japan. After 552 (this might be wrong), Buddhist priest and artisans start coming from the Asian mainland, and I'm pretty sure they were from Korea. So, that means the Mononobe might try to kill off the Buddhists who came in over the course of those three decades. Also, it seems that in 540, there were 7,053 households of the Hata clan already in Japan. The Hata were from what seems to be Silla, since I have doubts about Wikipedia saying they were from China or Baekje. So it would make sense to me that Buddhism probably came earlier than is recorded, and would still be present among the general populace, and I assume it would be concentrated among immigrants from Korea. That's why I'm not sure the Mononobe could get rid of Buddhism. After all, Buddhism reached every member of the Sinosphere.
I definitely could see a Nakatomi/Mononobe clash, though I could also see them working together if neither of them are able to get an advantage over the other.
However, there's the idea of centralization. I could see it, because I'm pretty sure the Taika Reform was caused by the rise of the Soga clan, though I could be wrong. Historically, the Soga were also involved in centralizing Japan too, so I think it could go either way, due to the pull of Chinese influence. I could keep looking at this.