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The short answer is no, neither Imperial Russia, nor USSR adopted segregation, due to distrust of non-Russians.
Longer answer: History of the Russian Imperial army can be divided in to the two major periods 1699-1874 and 1874-1917.
Peterian period (1699-1874)
Naturally a lot, changed in the period of 270 years, but foundational principles of Peter the Great were fallowed. During the Peterian period, force was raised mostly from Russians (that including Ukrainians and Belorussians) due to recruitment system being tied to serfdom: landholder were tasked to provide number of rekruts, from the serfs owned (the ratio fluctuated during the period, but in general we speak about 5-6 rekruts, from 1000 souls), hence the territories were manorial system did not dominated were outside of the recruitment system.
That by no means was some loophole that accidentally let minorities of the hook: it was intent Peters to have a Russian-majority army and system was designed around that, with no intent to draft non-Russian population. Often cited Imperial Decree of May 17,1798 (stating that recruits must be native born Russians) simply gave legal basis for long standing practice. Still, connecting recruitment with the manorial serfdom had unintended consequence of non-Russian serfs being drafted. In the first paper I linked bellow authors make assumption (page 4) that Mordvinians and Cheremisov Tartars fell under compulsory draft due to being considered more loyal and easier to integrate. Hardly the case, considering that these are the people that were eventual subjugated to serfdom and hence fell under rekrut system. Another-and better- example would be Lithuanian population, that also was subject of this system since the Third Partition, despite active participation in revolts against Russians, simply because serfdom was well established in Lithuania.
These non-Russian rekruts came from relative small ethnicities, easy absorbed and diluted in much larger body of the Russian servicemen, so no one raised much fuss that some of the serfs given to army were not proper Russians.Somewhat paradoxically, the officer corp was much more ethnically diverse due to inclusion of the nobility of the subjugated nations- Baltic German and Georgian nobility even made themselves a name by are providing a officers to the Imperial Army.
Also I should mention Cossacks- a rather interesting ethno-social group of the Russian Empire. Generally Cossacks were considered Russians, but large number of Tartars were absorbed in to the Hosts, and for all matter and purposes were considered Cossacks.
Aside from this "in the army by technicality", the only use of non-Russian troops was a relatively small auxiliary light cavalry force raised predominately from Muslim populations (Tartars, Bashkirs and such). As this force made only few percents of the overall standing army and were fighting in traditional manner (meaning that Russian empire was the only European power to field horse archers during Napoleonic wars), they never were of much importance.
Milyutin's Army (1874-1917)
As much as it matters to the topic we discussing, Milyutin's reform of the 1874 was targeted to update Russian military to the European model of mass levy army. The two major goals was to expand draft pool and establish mobilization reserve. To achieve that compulsory service for all males was introduced, while service time was cut from 12 to 6 years. That was on paper. In practice, the system had a lot of exceptions and fine print attached.
Most of Siberia (especially Siberian natives), Central Asia and Caucasus region were left exempt from the levy.Again, Curran and Ponomareff put all they attention on political reliability of the minorities, and that certainly was a large issue, but there is also some other practical considerations.
Take for example Siberia. While Russians lived here in permanent settlements, most natives like Mansi or Nenets were supporting themselves by hunting and elk-herding and continued to live semi-nomadic lives, interacting with outside word only by occasional visit of the market town. Given extremely low population of such tribal societies, drafting from Siberian tundra or isolated villages in Caucasus mountains would be simply a lot of effort for little to no gain.
Another problem was, in Milyutin own words "low level of civic development". Curran and Ponomareff quotes this to support statement that limited recruitment was adopted due to "distrust of some non- Slavic ethnic minorities" (page 5-6), but again, I would see here simple recognition of reality: if you grab someone from the ancestral yurt or aul and drop him in the modern town (or barracks) there will be some acclimatisation problems. Again- a lot of work for little profit.
Anyway, with large portions of the empire being politically unreliable or simply to hard to draft, disband of native auxiliary units was probably the largest change in the military as far as ethnic policy goes: just as before Army was still mostly raised in European part of the empire and still predominately Russian with other ethnicities being dispersed and diluted.
World War One and Revolution
Then war broke out, Empire found it self in want of more cannon-meat and began to weaponize national feelings of the various ethnic groups. Among the most notable examples are Armenians, eager to liberate historical Armenia from the Ottomans and Latvians, known (at the time) for long standing animosity towards Germans. Peterian- style Muslim cavalry auxilia also made return as the Savage Divisions.
However, that still remained rather the exception, a propaganda effort to get additional batch of volunteers. For the most part army remained uniform and Russian-dominated. Some units here and there had a larger group of non-Russians but more by accident rather than intent. For example Gen. S. Raštikis (he served in Caucasus Front during WWI) mentions few units with larger number of Lithuanians serving there.
That all began to change after the February revolution, then birds of feather began to flock together like no-ones business. Most prominent- and initially successful- were Ukrainians, by merit of establishing a government- Central Rada- as early as March. Not only new volunteer units (like Bohdan Khmelnytsky Regiment) were raised, but also a significant effort was made to assemble already serving Ukrainians in to national units. Pavlo Skoropadsky put himself on the political map by successfully reorganizing his 34th Army Corps into 1st Ukrainian Corps.
Polish officers in Petrograd formed a Chief Polish Military Committee (Naczelny Polski Komitet Wojskowy), who initiated formations of three Polish corpses ( Korpus Polski w Rosji )- one in Belurus, Second in Bessarabia and Third in Ukraine.
Other nationalities, without central institutions acted more from the local initiative and results vere mostly small lived small units. For example Lithuanian efforts amounted to separate battalion within the 1st Polish Corps, reserve battalion in Minsk, Separate Vytautas the Great battalion in Chelyabinsk , field hospital in Romanian front and dragoon detachment in Valka.
As Imperial army effectively ceased to exist in 1918 all these national units fell apart, were disbanded,became nucleus of national armies (like Ukrainians) or were simply dragged- sometimes against they own will- in to one or the other side of the Civil War.
To sum up Empire period it would correct to say that being composed mostly of uneducated peasantry (despite Milyutin's attempt to educate recruits) and dominated by single ethnic group, Imperial Army was capable to ignore dreaded "national question" altogether.
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Early Soviet Union
Formation of the Red Army was a mess. Suffice to say that Until March of 1919, Bolshevik leadership was still debating do they want to have an army at all. Hence RKKA had its begging in numerous units raised as revolutionary militia by by local Soviet or a gang of wannabe warlord or simply local self-defence force.
Due to such origins, many of the early Red Army units could by described as national units in nature, that was more of a product of the limited recruitment area and by no means desired by Bolshevik leadership.
The problem with national units was that they were more interested with national interest that world revolution. Even as Lenin founded a number of satellite states (Soviet Estonia, Latvia, Lithuania, Belarus, Ukraine etc.), the national forces of the said "Soviet Republics" had to be significantly augmented with Russian units, to counter nationalistic tendencies and to make them viable military force.
Even famous Latvian riflemen were not without problems. While many of them remained loyal to the Reds, a lot of Latvians used disintegration of the Russian Army to return to Latvia and joined national army afterwards.
Only "national units" that were unquestionably loyal were Chinese mercenaries. As long as they were paid, that is.
To sum up experience of the civil War, Soviets had some limited success with national units in cases like Latvians, but overall they showed themselves untrustworthy and unreliable and it is generally bad idea to organize your army from what effectively is your main enemy. While Kolchiak and Denikin and other major Whites were dealt in a matter of the few years, Red Army's main task for the next few decades remained stomping out uprisings that were constantly popping out across country.
In such situation, national units were out of the question and falling back on Czarist principle of unified Russian dominated army was the only course Bolsheviks realistically could have taken, hence in the 1920 over 95% of the standing RKKA was Russian.
However experiments with national formations continued to exist up until 1938.
Georgia, with population of aprox 3 000 000 was permitted national force of two divisions, half the size of Ukraine's (aprox. population at the time was 30 000 000) and twice the Belarus that had to do with one division for 11 000 000 people. However Belarus did not had Borotbists/Ukapists like Ukraine or Mensheviks like Georgia.
Despite all the fancy plans to expand these forces and use them as a training ground for specialized units, national forces were kept at bare minimum, making sure, that it will newer be able to challenge to RKKA- proper.
In the early 1920's they existed because Moscow still did not had enough control to ignore or silence opposition demands in Ukraine and Georgia, but as time marched on and Party's control strengthened, they become and more just something that, well just exist. Part of why they existed for so long was that it was part of the set of policies, known collectively as Korenizatsiya. The general goal was to attract non-Russians by appealing to cultural needs and eventual to grew communist cadre in non-Russian population. By 1938 Stalin put the final nails in to the coffin of this Korenizatsiya: Russian nation was proclaimed "the Big Brother" over the rest, Russian language become mandatory and national units (such as they were) we disbanded.
Second World War
The closest thing to national units Soviet army had in 1941 was 3 Territorial Rifle Corps (22th, 24th and 29th) formed from the national armies of the recently occupied Lithuania, Latvia and Estonia. Needless to say, despite all the reeducation and purge efforts these units were far from reliable and from the first day of the war soldiers of the troops of the said Territorial Rifle Corps distinguished only by desertion and avoidance to engage or openly siding enemy (albeit distinguished might be to strong of a word, given that in summer of 1941 such behavior was endemic across entire Red Army).
the first actual national unit raised during the war was 201th Latvian Rifle Division (renamed 43th Guard after the Battle of Moskow), raised in August-November of 1941. Others soon fallowed and in total there was 14 rifle divisions, 15 rifle brigades and 20 cavalry division raised as national units.
This was done, despite inherent untrustworthiness of national, first of all due to logistics- these national units were not raised by RKKA but by local Party organizations with the Party resources. Essentially, assigning additional recruitment to the Party institutions permitted to field additional units without putting additional logistical pressure on the RKKA. And given, how Soviets conducted the war, the need for fresh cannon-meat was constant.
Also, these units served a propaganda role (especially Baltic ones).
Now, one thing to hold in mind, that being organized as a national unit, does not mean that it was such unit in the literal sense of the term. New reinforcements would come from whatever army had at the time (and often quite purposely unit would be reinforced with other nationalities) so even if titular nationality was dominant in the first place, it would not last.
For example 16th Lithuanian Rifle Division as of April of 1942 had 8129 soldiers and officers of whom 3 923 were Lithuanian. That is numbers then division was few months in to formation, before the deployment. Numbers only went down afterwards, and by the time of first engagement Lithuanians made up roughly 1/3 of the division*.
So in this case division was hardly "Lithuanian" to begin with and as security apparatus was constantly reporting political unreliability and dissatisfaction of the Lithuanian troops, Soviet leadership was more than happy to keep it that way.
Post War
Well, thats a short story. As national units, by all measure was more trouble than they were worth and Party has no longer had any regional opposition to appeal to, this units were disbanded. Some just after the war (249th Estonian Rifle Division was disbanded in 1945, 308th Latvian rifle in 1946) some still existed in garrison duties for a time (16th Lithuanian Rifle and 43th Guard were disbanded in 1956). But the rest of the duration of the Soviet Union and Army, soldiers served in uniform, multi-ethnic units with all the problems that produced.
* 16th Division is a bit unique due to the fact almost from the beginning it become know as "Jewish division" and in fact had a largest percentage of Jewish personnel (about 1/3).
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Sources:
Susan L. Curran,Dmitry Ponomareff. Managing the ethnic factor in the Russian and Soviet armed forces
Juliette Cadiot. Russian army, non-Russians, non-Slavs, non-Orthodox: the risky construction of a multiethnic army. Russia, USSR
Rimantas Zizas. Raudonosios armijos 16-oji Lietuviškoji Šaulių Divizija: kelia nuo susiformavimo iki katastrofos prie Aleksejevkos (1941- 1943 m. vasaris - kovas) in Karo archyvas XXII pages 291-352