In the US Physician licensure is on a state by state basis with some exceptions for federal facilities (like VAMC, IHS, etc) where any state license can be used.
Some states do grant temporary licenses, volunteer licenses, and or camp licenses and it is possible that Nassar had something like that during his time at the ranch. It is also possible that USAG somehow overlooked this entirely.
Also, since he went there so often, he may have obtained a Texas license. It's not hard, mostly the same as getting a new driver's license when moving states but more expensive and some extra paperwork (CV, license from previous state, background check). My sister was originally licensed in only Minnesota, but then got hired by a hospital system that overlaps with North Dakota, and since they rotate through facilities in both states she had to get a ND license too. Putzy and time consuming, but if you travel frequently it would be easy to obtain. Especially with endorsement from a USOTC and the money of USAG available.
Wild speculation though, I have no idea what they did in re licensure.
....I just typed license so many times it's starting to look weird and wrong. Woo!