That said, the NICS system is managed and run by the FBI, so I can't speak authoritatively there. In dealing with them regarding it though, the impression I got from their tech team was that it wasn't storing anything past receipt acknowledgement, and even where it stored data, the implication was that it only stored the metadata there as well, uniquely identified by the NICS transaction ID.
If true, clearly someone would have to be holding NICS records and the ATF would have to have access to them.
There might be, and probably is (at least) a record of the transactions that were processed and the result of the transaction. Also, as I understand it, if the check fails (hence, no gun is disbursed), then the PII may be held longer to aid potential investigations.
From the ATF's perspective though, PII definitely isn't kept, or at least wasn't as of 6 years ago. I honestly don't remember the specs on failed checks. I don't think that it did, but that isn't to say that if they are prepping for an investigation of an FFL that they couldn't requery the FBI for more thorough NICS records than what they hold in their own systems.
That's good information on the NICS records. It would seem, though, that given the information in the NICS check record and information attainable from a mandatory bound book inspection that the end result would be the personally identifying information of the transferee. It is good, at least, that this would necessarily be a manual process.
I don't have any personal insight into what happens to them after that, but it seems like at least a sparse, time-delayed registry could be built from those records... which is perhaps the best kind of registry as private sales would skew the data substantially.