How is the concurrent tracking of "state position" in NFA's typically modelled? Conceptually, to track the multitude of "state positions" arising in an NFA while it is processing a given input string, a quantitative notion of memory/cache/storage is required per "fork" of execution over the automaton, for keeping track of the states where the computation is currently "at".

What is the typical theoretical treatment, regarding the complexity of this degree of concurrency?!
The theoretical treatment is just to say "It's what happens: that's what nondeterminism is." The machine accepts if there's any computational path that finishes in an accepting state.

Anything about memory, caches or concurrency is a matter of implementation.  The simplest implementation is to keep a vector of Booleans that records whether or not the machine "could be" in each of the states at the current step of the computation.