Alliance
The Night Elf nightsaber mounts would probably not be able to carry heavy plated warriors, at least not for any practical length of time for a military campaign or long distance travel. Even if we are generous and compare Azeroth's nightsabers to our world's ligers (the biggest cats), they still do not have the same bone structure or hooves as horses to support a lot of weight on their backs for long periods of time. So the nightsabers would be probably limited to carrying light, unarmored elf chicks. The biggest selling point of the nightsabers would be for ambushes. The nightsabers are cats, and are thus naturally inclined to be stealthy. Any environment with a lot of wood or bush is going to give saber riders a good possibility of launching an ambush. The saber would also have excellent night vision too. Metal armor would clink and gave away the game, so again the rider is probably going to be lightly or unarmored. Cats are among the fastest land sprinters it would be very difficult to escape an ambush. They would also be able to climb up trees, perhaps even launching a strike from above. Disadvantage would be that they would need to eat a lot of meat, unless the Gnomes have invented cat food.
Gnomish mechanostriders would be among the best modes of transport in WoW. They are not living beasts that need to be fed or can become temperamental or need to be trained or need sleep or can die of heat or cold or can be killed by arrows. They are made of solid metal and are hard to destroy, and only require fuel. Given how industrial the Alliance is and how they have
a modern day logistics system, the world's largest mercantile fleet and a robust air force and the ubiquity of Alliance supply depots stocked with engineers and parts, maintaining and fueling the mechanostriders should not be an issue. The mechanostriders and other machine mounts wouldn't be useable in every single situation; I'd imagine they would be quite noisy so nightsabers would still be better for ambushes, but they would be superior to most mounts.
Draenei Elekks = Indian War elephants. Would be best used as platforms for archers to stand on and shoot from during pitched battles, or to pull heavy loads. Not very practical outside of that as they are not as fast or as agile compared to other mounts and are food guzzlers.
Horde
Horde Wyverns are also cat-like, so they might be good at stealth and ambushes. Their cat-like eyesight should also be good for spotting in the dark.
Tauren Kodos would be great at pulling wagons. Doubt it would make for a good war mount, though. Sure, ingame all mounts run at the same speed, but looking at a Kodo I can't imagine it running at 40 miles per hour or turning on a dime. Same issue with Clefthooves, but at least those have tusks and are huge so they can plausibly be used like Indian war elephants, being used to demolish fragile wooden or mud huts, a platform on their back for archers to shoot from, and sheer intimidation factor.
Troll Raptors could make as good mounts for scouts who need to traverse rough terrain, but I do not see them as being fast sprinters like cats nor as good long distance mounts like horses.
Skeletal horses presumably do not need food, and probably won't fall over they are hit by an hour. Perhaps they do not even tire. They would be rather machine like, perhaps even better than machines since they wouldn't even need spare parts of fuel to keep them maintained. Skeletal horses could be perhaps the absolute best mount to use in the setting. Only issue is that they might be very foul smelling, though that could be mitigated through the use of incense and perfumes.
I am skeptical that Hawkstriders would make for good war mounts. If they are like our birds, then they would have hollow and fragile bone structure. Doubt they could carry a heavily armored warrior for long.
Goblins can create automobiles and bikes, which should be superior to most beasts as transports as the machines cannot become exhausted, do not need to be trained, do not die from a lack of food, cannot die of heat or cold, etc. Unfortunately the Horde is not depicted as having a vast industrial-logistical apparatus, so maintaining a constant supply chain of parts and fuel for Horde forces across the globe would be very difficult. Also, there is the issue that Goblin engineering is shoddy, prone to breakdowns and sometimes even harming their operators, which would make them unreliable as army issued transports.
Zandalari Direhorns would be fantastic for pitched battles. They have tusks like elephants and clefthooves, but unlike them their faces are very bony shields, meaning they can charge into battle and are unlikely to get their face smashed. The eyes are vulnerable but can be protected by barding. I don't see them having much application outside of that, as they wouldn't be as fast/efficient as horses.
Brutosaurs could make for great platforms to for archers to stand on and shoot from, like war elephants.
Other
Pandaren turtles: similar issue to Kodos where I just can't see them being fast or agile enough, and the turtle mounts depicted ingame aren't quite large enough to pull wagons or build platforms to stand on and shoot from. Probably the most impractical racial mount.
Felsabers would inherit most of the Nightsaber's attributes, but do not appear to be as big, so perhaps they could not carry as much weight. Felsabers appear to have rougher, slightly armored skin, which might make them less likely to go down in battle. Their mane of green Fel fire might make them unfeasible to use by normal people who cannot negate Fel fire.
Drakes would be among the best mounts to use in the setting given that they can fly and their great strength, but they would be very costly to feed in terms of meat. The Alliance already has many, many airships and gyrocopters to transport things by air, which are more efficient than feeding Drakes. Drakes would probably be more used by the Horde given that they do not have as many ships or airships as the Alliance, and their small Wyverns can't carry Tauren or heavy loads.