While American Indians were "granted" citizenship through a series of acts during the interwar period, it was very much a second-class citizenship, with a number of states still denying them the basic rights we associate with citizenship such as voting. And indeed, I place granted in quotations above because it must be remembered that citizenship was thrust upon the native communities regardless of whether they wanted it, and the changed it would bring in their relationship to the Federal government, or not. I'm not a scholar of native communities though, so only am ready to speak to conscription, so hopefully someone can expand on the deeper, underlying aspects there. The specifics aren't too important for the core answer here though, beyond understanding it was a fraught relationship (for more on the aside here, I'd recommend this answer by /u/snapshot52, tracing the path to citizenship).

Anyways though, from here we need to backtrack slightly since while you are correct in a strict, legal sense that there ought to have been an exemption prior to the change in citizenship status, during the First World War this was often simply ignored or elided over. The Eastern Band of Cherokee, for instance, were overseen by Indian Office (now BIA) superintendent James Henderson, who, despite knowing full-well they were exempt, simply instructed the young men of the reservation in western North Carolina to register for selective service, even threatening to arrest those who refused, resulting in nearly forty men getting drafted, plus more choosing to enlist ahead of a likely draft. Henderson was apparently motivated by several factors, including personal - the hope of getting some of the notable troublemakers out of his hair, (misguided) moral - believing that service would help "civilize", and organizational pressure - Bureau leadership wanting the Indian office to be seen "as a responsible and patriotic part of the Wilson Administration". Nominally this meant encouraging voluntary enlistment, but as seen here, that was not always the case.

It also ought to be noted that Henderson was hardly one, single rogue Indian agent, with similar attempts to force the native, non-citizens onto the Selective Service rolls on other reservations, resulting at times in both passive and active resistance, although generally on a small scale, and put down quickly by local (white) militias or vigilantes. The BIA essentially passed on their duty in such cases, doing nothing to rein in such acts or provide legal clarity, as the issue did require at least some nuance, since some Native Americans did possess citizenship, meaning it was more than an automatic "No". This gets to your specific question, as it meant that the Native American conscripts were essentially treated as any other local draftee, counting towards the conscription quota for Swain County, in the case of the Eastern Band, where they resided. And for those few who were dogged enough in disbelieving their eligibility, their case was nominally left up to the local draft board to determine eligibility, something which they were quite unprepared to handle, and resulting in a variety of outcomes.

In more than a few cases, we can likely surmise that draft board didn't particularly care for the legal correctness, and were just happy to use native bodies to fill out the draft quota for their county (and this of course also intersects with white imaginations of Indians as 'great warriors' so many considered it the proper role for them to play anyways). For large reservations, where there were (believed to be) some reasonable number in the population who did possess citizenship and thus draft eligible, induction personnel were sent to set up draft stations on the reservation (although, best I can tell, it still would have ultimately been answerable to the local regular draft board), but these were met with suspicion at best, and at least some required federal officers to accompany them for protection. In simplest terms, even if some residents were citizens and could be drafted legally by such boards, locals simply didn't trust them to be careful verifying such eligibility.

Now returning to World War II, the citizenship issue was essentially settled, at least by strict reading of the law, so while the phenomenon seen in the First World War was not a universal one, being dependent on the honesty of the local agent, and at least in theory, an individual persistent enough could sometimes find legal remedy, there was no longer any such protection against conscription in the Second World War, but that didn't mean there wasn't still opposition to the draft. To be sure, compliance with Selective Service requirements were generally followed, and in the case of the Eastern Cherokee at least, there were no attempts to avoid registration.

However, returning again to your core question, as in the First World War, the draftees were overseen by whatever local draft board they fell under in their county of residence. This was not without controversy though! As noted earlier, at least some cases in World War I where a draftee's appeal based on non-citizenship was ignored, it likely was to help the county meet its quota, and while generally willing to serve, prospective draftees from the reservation had no particular interest in helping a local board meet its quota. To this end, the Eastern Band council issued a Resolution on Nov. 5th, 1940, specifically requesting to be allocated their own draft board independent from those of the local counties that their reservation intersected:

It is not that we wish to shield our young men and prevent their being inducted into the Service. It is not that our young men wish to shirk any duty which may come their way under the draft... And we say here and now, if we may do so without seeming to boast, that there is not a more patriotic people to be found anywhere in this country of ours than the Cherokees of North Carolina... We feel that the ends of justice will be better served so far as this reservation is concerned if a fair and impartial draft board is set up here, separate and apart from these county boards. We feel sure that a board thus set up would act understandingly and in a sympathetic manner in the induction of our young men into the service

The request was not granted, but it is also worth noting that many young men enlisted voluntarily, for a variety of reasons. Finger notes that through mid-1942, enlistees had had outpaced draftees from the Eastern Cherokee, and by the end of the war - during which voluntary enlistment was only allowed on a very limited basis - it was still 123 to 198, respectively. But while native communities provided a noticeably disproportionate number of soldiers compared to their total population, with 25,000 or so serving during World War II, that doesn't mean they were unwavering boosters either. Some argued for Conscientious Objector status based on their religious beliefs. Others believed that whatever their nominal citizenship status with the United States might be, they were nevertheless members of their own nation as well, with treaties in place with the US government that they argued made them exempt from the draft.

And as noted at the beginning, despite citizenship, indigenous people were still denied some rights that ought to come with it. There is an interesting mirror there in how, during World War I, the draft was forced upon some despite their lack of citizenship; while then during World War II citizenship had been forced on them and thus the draft was now legal, so while the former were trying to argue that they weren't citizens, with the latter it helped energize activism to argue for the full rights of citizenship. If they were being drafted due to their citizenship, ought they not have its rights protected, such as voting? Although early attempts to assert it, and register to vote, were denied, military veterans returned as some of the strongest advocates in the immediate post-war years for full access of their rights denied.

So this obviously ended up covering a bit more than you asked, but hopefully it brings home that the question isn't quite as simple as the title. To be sure, with regards to the very specific arrangement of the draft-boards, it isn't all that complicated, with the drafting of Native Americans run with no real difference from anyone else, and handled by the same entities, even if this wasn't always agreeable. But the nature of citizenship and the American Indian was a little more complicated, especially when taking a slightly broader view of both World Wars, as conscription didn't always align with citizenship, even if it ought to have, and citizenship did not always align with the rights that it ought to have conferred either.

Sources

Carroll, Al. Medicine Bags and Dog Tags: American Indian Veterans from Colonial Times to the Second Iraq War. University of Nebraska Press, 2008.

Finger, John R. Cherokee Americans: The Eastern Band of Cherokees in the Twentieth Century, University of Nebraska Press, 1991.

Finger, John R. “Conscription, Citizenship, and ‘Civilization’: World War I and the Eastern Band of Cherokee.” The North Carolina Historical Review 63, no. 3 (1986): 283–308.

Tate, Michael L. “From Scout to Doughboy: The National Debate over Integrating American Indians into the Military, 1891-1918.” The Western Historical Quarterly 17, no. 4 (1986): 417–37.

Zissu, Erik M. “Conscription, Sovereignty, and Land: American Indian Resistance during World War I.” Pacific Historical Review 64, no. 4 (1995): 537–66.