First point, is it generally agreed that preventing the rise of Islam* would lead to significantly less demand for slaves over the next millenia or so, specifically from Eastern Europe and East Africa?
Less demand? Not exactly : we're rather talking less offer, and less demand arising from the possibilities. Meaning that, with the right opportunities and with a slavery that didn't disappeared from pre-Islamic Mediteranean basin and Near-East (Asoristan, meaning roughly Mesopotamia/Iraqwas a region that particularily resorted to productive slavery and semi-slavery for exemple; and domestic slavery as used by Arabo-Persians have its roots there), it could boom as IOTL.
Both Romans (and Barbarians with them) and Sassanids had a technical rule about their own coreligionaries not being enslavables. It was generally passed trough by enslaving neighbours, considering recently converted population conveniently not that converted, or flat-out ignoring rules and going with whatever rolls.
Now, you'd have changes : the Persic Gulf tended to replace the Read Sea as the main trade road to Eastern Africa and Indian Ocean (which were servile pools since the turn of the millenium, growingly so), playing a regional role, but possibly a redistributive role up to China. But it's certain that without a same legal/economical continuum in the region, Sassanids would be the main beneficiaries (keeping in mind that the Romano-Persian wars were the occasion for them to capture a large part of European slaves out of defeated armies).
it's hard to really give numbers for how much slaves were taken up to the Xth century, but while it was high (to the point Zanj went to mean Black) it was maybe not this high (altough the systematical practice of castration asked for a large renew, you did have a "natural reproduction" of the chattel). Personally, I'd expect an even more important Sassanid slave trade in Eastern Africa, giving Western Africa and Central Europe would be closed. You could make the argument that Central Asia could become more of a pool for slavery (altough more domestic based than in Africa) ITTL.
The most critical changes would be in Western Europe and even more in Western Africa.
Not that Europe didn't knew slavery : Franks and Romans practiced it, but while Romans maintained its use on a significant scale up to the Xth and XIth (mostly from Black Sea IRRC), Franks seems to have practiced a much more blurry distinction between servants, clients and slaves (which would eventually give birth to serves), especially in the northern regions (arguably Mediterranean regions had much more socially distinct slaves, as in Spain and Italy), and giving that the neighbours you could raid for enslavement were (at least technically) subordinated, you couldn't really do that in significant numbers.
IOTL, Franks harboured a revival of slave trade in the IXth, mostly in order to ask for the demand in al-Andalus, essentially with Slavs (which gave the name to both slave and saqaliba, both meaning the same). ITTL, without al-Andalus, would the demands from Gothic Spain and southern Gaul be enough? I frankly doubt it (especially as no Islam means no crisis of the mid VIIth or a less serious one, and then likely no violent policy of re-affirmation of power over Germanic principalties, etc.).
One could argue that if Merovingians manage to hold off against Slavs ITTL, you might have an offer that no matter what, would have to be shoved to someone, but Romans make better candidate for that IMO, and they already have their own markets.
So, while I don't see slavery disappearing earlier from this PoD from Western Europe (and possibly later, as a consequence of not having to deal with the collapse of Carolingia), I don't think it would be that important as an economical and social development (and of course, it means no Abassid crisis, no repercussion in Scandinavia, likely no Vinkings raids or at least not as important, etc.)
Politically, it could mean a lesser development of the earlier Slavic principalties on the Baltic, as their control (or tribute) on slave trade was part of their sophistication process.
It's even more clear for Western Africa : nobody really gave a **** in the Mediterranean basin about what happened there. Sure, Ghana slowly structurated itself, but you really had to wait the the Arabo-Islamic conquest IOTL to have the first permanent and dynamic trade roads to the Niger. ITTL, it could mean a slower development, as neither Goths could, or Africano-Roman wanted, or Africano-Berber able to have a go in the region for the immediate aftermath. The dynamic in Northern-Western Africa was at best South-East leaning trough Libya and Darfur.
