Command: Imperials, Dunmer, Redguards, Bretons High command (Field Marshals, high-ranking generals): Imperials, perhaps Bretons Mid-level command (Brigadier generals through Majors): Bretons, Dunmer, Redguards Officers and NCOs: Imperials, Redguards
Heavy Infantry: Orsimer, supplemented w/ Imperials
Scouts: Argonians and Khajit
Skirmishers: Bosmer, Nords, Redguard supplemented w/ Khajit and Argonians
Line Infantry/General Infantry: Redguards, Nords, Imperials
Mage Corps: Dunmer, Altmer, Bretons
Medical Corps: Bretons, possibly supplemented by Imperials
Logistics Corps: Orsimer, Khajit, Imperials, possibly Nords
Artillery: Dunmer, Redguard
Cavalry: Imperials
Navy: Imperials, Nords, Dunmer, Altmer, possibly Argonians
Intel Corps: Khajit and Imperials (Khajit spies, Imperials for counterintelligence and command)
Engineering Corps: Imperials, Altmer supplemented by Orsimer
Reasoning:
Command: Imperial's charisma, education, intellect, and military experience makes them ideal for officers and commanders. The intellect and structure of Breton society also makes them ideal for mid-level command, albiet not as good as Imperials at frontline command. The cleverness and military experience of Redguards and Dunmer makes them excellent at brainstorming, planning, and asymmetrical warfare
Heavy Infantry: Its pretty obvious why I choose Orcs. Their affinity with heavy armor, block, and both one-handed and two-handed weapons makes them excellent shock infantry. However, their low total population led me to supplement them with Imperials, who, while not quite as good, are still excellent in heavy armor, block, and one-handed, and their skill in Restoration may also come in handy
Scouts: Argonians here are an obvious choice. Due to their high sneak, light armor skills, ability to breathe underwater, and immunity to disease they are unaffected by most obstacles, enabling them to take unconventional routes to reach advantageous positions. Khajit, while lacking the versatility of Argonians, are excellent at sneaking in and finding information, making them both excellent scouts and spies
Skirmishers: Skirmishers, similar to scouts, need to be quick and light. However, they also have the job of harrassing the enemy. The light armor and archery skills of the Bosmer make them a superb choice, with Redguard's stamina and archery making them an excellent choice as well. Khajit, though less ideal than the previous two, are still excellent skirmishers, as are Argonians, while Nord's ability to withstand adverse weather conditions and skill with light armor and two-handed weapons makes them excellent for raiding parties.
General Infantry: Imperials are great all-rounders, making them excellent medium and heavy infantry. They also have good command capabilites, as I mentioned. Nords and Redguards are born to fight, and are very good at close range assaults while remaining mobile thanks to their skill in light armor.
Mage Corps: Bretons, Dunmer, and especially Altmer are the greatest mages in Tamriel. Their aptitude for offensive magic makes them excellent at supporting infantry from a distance. While Bretons may lack the firepower (no pun intended) of the elven races I mentioned, their aptitude in conjuration makes them extremely versatile
Medical Corps: Bretons here are an obvious choice, due to their skill in both Restoration and Alchemy, which makes them great at both surgery and producing medicines and potions. Imperials' skill in restoration makes them excellent field medics and surgeons, despite their lack of alchemical talent
Logistics: This one is also a field where Imperials, due to their intellect and leadership skills, excell. They are great at organizing and managing supplies. The smithing and enchanting abilites of Orsimer make them ideal quartermasters, able to easily manufacture, upgrade, and repair equipment. Lastly, the long history of the Khajit and their caravans makes them superb at transportaion and bartering, enabling them to transport tons of supplies over vast distances in relatively short order
Cavalry: While cavalry don't play a huge role in TES, the Imperials are probably the best suited to this.
Navy: Imperials, Nords, and Dunmer likely have some of the best sailing experience, at least from what we can see in the games. While I'd bet that the Altmer are also excellent sailors due to them living on an island, I'm not entirely sure of this so I won't jump the gun. Nords are also excellent marines, able to quickly disembark to attack enemy ships before returning. Dunmer and Altmer, with their aptitude for destruction magic, would be able to burn down the sails and rigging of enemy vessels, imobilizing them. Argonian's immunity to disease and ability to breathe underwater might make them good sailors, but I have no knowledge of their actual sailing experience.
Intelligence Corps: Imperials and Khajit are the obvious choices here. Imperial education and talent for oration makes them excellent at counterintelligence, interigation, and sorting out information. Khajit's experience in travel and superb sneak skill makes them excellent at locating and stealing enemy information, as well as sabatoge, and their claws allow them to fight without the use of a weapon, perfect for undercover operations.
Engineering Corps: Now we get to the final component. The talent possessed by Altmer and Imperials in terms of architecture and engineering makes them excellent at building and fortifying structures. While Orsimer may lack this understanding of architecture, their aptitude in smithing allows them to easily design and manufacture components required for building.