The number of firearms that are discovered that dropped off the grid just due to forgetting where the papers are is actually more common than people think.
Remember the lady who tried to turn in a STG-44 in the buyback? The officers helpfully told her to sell that valuable piece of history for something more than the gift cards they were authorized. Checking the NFA registry is actually a fairly straightforward thing to check.
The ATF will check their records and determine, based on all of the markings available find whether or not it on the NFA registry. The complication comes into it because older firearms had a wide variety of markings and the markings the ATF know are only the ones put on the registry page. Names, cities, all the available information to check against typos and other human error in the registry.