I am aware that substantial parts of the Golden Horde were Muslim, but those as far as I remember were the Golden Horde's Transoxiana territories and Volga Bulgaria. Kypchaks or Cumans were not predominantly abrahamic as far I can remember. Regarding the Ilkhanate I read that despite it converting to Islam, they still were waging wars on their coreligionists, so diplomatically it may not change a lot. But I admit I am not an expert on the subject of demographics myself. I had only read limited sources. If you have any information on topic I would gladly read it. Still I think that at that time the demographically dominant Turkic nomads had a quite diverse religious demographics, so if one Khan decided to forcefully enforce one creed over the other than it may shift the dynamics at least in the steppe areas.
I don't have any specific figures, just conjecture from a few points.
A) He managed to win the the internal military strife against the pagan (using to describe anything non abrahamic in this context) aristocracy that were quite numerous, suggesting a large basis of support from populations.
B) I wasn't just talking about Muslims, but particularly Christians. Whilst the rural and nomadic population were a strong mix (and hard to document properly), entitites like the city states of the Rus etc were primarily Abrahamic.
C) Whilst there is a degree of Paganism amongst the Turks at this time, quantifying what that meant is challenging. Syncretic views were quite popular with northern turks in general throughout the period, and somebody have been described as pagan despite following Christ or Muhammad. I expect in time, just as OTL, these syncretic views on the Abrahamic faiths would have continued to give way to some form of Orthodoxy.
Now that isn't to say that things woudl have gone poorly, but that it is worth considering. The idea of the turks being strong manned into buddhism in the long turn is an interesting concept, although I don't think it is an inevitablity of such a change, demographics do tend to dominate policy and I am inclined to think that eventually the Golden Horde would have converted to Christianity or Islam either way.
Regarding the Ilkhanate, there is more of a complicated history there. Whilst there was war with co-religionists, they had a particular ire for the Golden Horde already. Giving them yet another excuse doesn't make war inevitable, but it does make it more likely.