This is an intriguing discussion about donkeys and onagers, and it´s become especially valuable with the added input of chariotry / asinary speculations.
Here are my two cents:
I think donkeys COULD potentially be selectively bred for greater speed, we cannot absolutely exclude that, but I doubt that they WOULD. The two big differences between our world and the no-horse world are in 1) what the animals` natural behavior inspires humans to do and 2) which environments they come from.
As for 1), the horses` natural speed as compared to the donkeys` natural persistence had a lot to do with why humans invented horse-based warfare and not donkey-based warfare, even though OTL`s Ancient Middle East had a lot of wars throughout millennia in which they had donkeys and onagers but not yet horses (3th and early 2nd millennium BCE). Horses entered the military world of the Fertile Crescent and Egypt around the middle of the 2nd millennium BCE, primarily in the form of chariot-using Hittites and Hyksos. (Chariot warfare had probably been used farther North for several centuries, while nothing comparable evolved in the Fertile Crescent.) All the speculations about war donkeys ought to take this into account.
Also 2), horses as mounts as well as as chariot-pulling animals developed on the Eurasian steppe, where the vast spaces which herders had to cover inspired mobility-oriented solutions. In Mesopotamia, Egypt and the like, people would look in animals for solutions to ease their hard labour. They did so throughout the 4th and 3rd millennia BCE. I don`t yet see why this would change in TTL´s 2nd millennium BCE, in the absence of charioteer arrivals.
Therefore, I would think
They'll follow if they get overrun by donkey chariots Hyksos style.
The sequence of events for the adoption of donkeys as cavalry mounts, as I see it, goes as follows:
East Africans domesticate donkeys and Sumerians use onagers as chariot animals as OTL.
Sumerians acquire donkeys by trade, and realize their potential usefulness as chariot pullers.
Sumerian war donkeys quickly supplant Onagers due to being easier to train.
Use of donkeys encourages development of chariots as archery platforms and battle taxis rather than as shock weapons, since donkeys have a stronger sense of self preservation than horses.
Donkeys are bred for size and strength to pull larger chariots.
Chariot crews fighting on rough ground improvise and start directly riding their mega-donkeys, perhaps inspired by seeing children do the same with other beasts of burden.
Chariots gradually fall out of use as it is realized Asinary are more efficient.
Ta da, Donkey cavalry in a logical cultural and military context.
Who would come to overrun anyone with donkey chariots Hyksos style? I think it would be biologically perfectly possible, but is it also historically likely to occur at all?
My alternate sequence of events would be as follows:
East Africans domesticate donkeys and Sumerians use onagers as chariot animals as OTL.
Egyptians use donkeys as chariot animals as OTL.
Stronger (but not faster) donkeys are bred under these circumstances.
Donkeys spread Eastwards across the Levante as OTL.
In warfare, stronger donkeys are used, like oxen, to pull wagons with equipment, foodstuff etc., and maybe to pull chariots in which high-ranking commanders are comfortably carried to the battlefield site instead of having to walk like Jack Infanterist.
Donkeys spread farther into Europe, where they are bred to support colder climate, and into Western Eurasia, where they are competing with the Bactrian Camel as beasts of burden, though.
On the steppe, they are more frequently used as mounts, including in military contexts, but not for speedy hit-and-run and shock tactics, and instead maybe, in addition to cart-pulling, also for the elevated position riders have. Here - if anywhere - more speedy donkeys would be of great use, and if it all, they`d be bred for this purpose here. In all likelihood, beginning no earlier than the 1st millennium BCE, though.
