So, what about the Indo-European migrations? How do they develop ITTL?
Sort of just fleshing out my earlier answer in a way that touches more on IE:
As I understand it, I think you'd still see some expansion that parallels early Indo-European movements probably at least across most of the Western+Central steppe zone once the wheel is invented, and wagons and carts become possible, with or without the horse. Those people could be early Indo-European speakers, of a sort, in an ATL (though they need not be).
In our timeline, the people the current majority hypothesis suspects to be the early expansions of the proto-Indo European speakers are taken to be the Yamnaya culture (either this or a very slightly earlier archaeological phase). In simplified terms I think the idea is this seems to from form earlier groups who are mixed hunters, fishers and herders who have begun to domesticate the horse, once the wheel is invented and then wagons which allow them to become more mobile and independent from the camps that they are tied to, and from the river valleys that are sources of water and fisheries. It seems like, based on archaeogenetics, then pretty quickly after you have expansions all the way to the edge of the West-Central steppe*, in the form of the Afanasievo culture who seem genetically identical to Yamnaya, once this happens, and you also have the expansions into Europe (via the Corded Ware culture and probably other movements).
It does seem to me like the wheel and the presence of lots of domesticated herding animals, is the pretty decisive factor in this mobility, rather than the horse. We know that at the Botai in Kazakhstan, people seem to have domesticated the horse as their only domesticated animal** - judging by archaeological evidence of milking, corraling, riding - and not have any of wheel or domesticated sheep, goats or cattle. They seem to have lived in villages and permanent settlements, not to be highly mobile and certainly don't seem to have expanded far and wide. Likewise, there is some evidence (less strong) that early cultures on the western steppe at Khvalynsk taken to be of pre-proto-Indo Europeans who were experimenting with herding also had domesticated horses. But again they don't seem to have really expanded in the big way that the Yamnaya did (though probably some interactions with cultures in Eastern Europe outside the steppe zone).
Hence I'd say there's probably a fairly good chance of some sort of wagon using herding based expansion by people speaking a language that could have had similarities and relatedness to pIE. Though this is still rather very chancey - probably quite a few languages spoken on the Western steppe alone, and other people in SE Europe and Caucasus at least were experimenting with the wheel (and might more successfully expand into the steppes).
Now though I don't think the horse was crucial to much of the expansion (stressing this is just my impression), I must be clear though, that it probably did have a major effect in making pIE groups more effective. In herding large herds of cattle and sheep, as a domesticated meat/milk animal in its own right (very well adapted to cold, arid grasslands esp. over winter compared to cattle and sheep), scouting around on horseback (to find good pastures, etc.), military raiding, etc. This all also has feedbacks into culture beyond the immediate practical applications. So even if you had a "no horse" wagon based pIE expansion of sorts, it would probably have been weaker in some senses, and maybe less successful in conflicts, and their language and religion may not have displaced as many others.
*Although fascinatingly it seems, not further than the West-Central steppe. It seems that this was the limit of Yamnaya-Afanasievo Indo-European expansion, and genetic influence stalls out at moving further until much later (over a thousand years later) with late Bronze Age Western-Central steppe cultures (though to be proto-Indo-Aryan speaking / "Satem") begin to interact with Eastern steppe populations.
The Eastern steppe populations seem to have an independent genesis, receiving domestic animals via the Inner Asian Mountain Corridor or from the Western-Central steppe, but not many people (because not many genetic signals) possibly even no people and most probably no language.
This probably has something to do with why IE influence (in the sense of the early genetic ancestry + linguistics) in Eastern steppes remains limited today, as where it happened it probably involved of later movements in a region that already had sophisticated herding economies among groups that spoke different language families.
Also raises interesting questions for theories that early IE groups represented by Afanasievo crossed over the mountain boundaries along the IAMC and Tian shan into the Tarim basin and were responsible for later Tocharian groups, and theories that very early IE groups had influences on early Sinitic groups.
**There's a question about whether Botai culture actually bred horses in captivity and selected them for traits and raised them for meat, or just tamed wild horses, but certainly they did use them to riding and milking it seems, and otherwise lived in settled villages with pottery and neolithic toolkit.
