While at least some states cross reference drivers license and CWP lists so that an officer running a DL check is notified that the driver has a CWP there is no permanent list kept of background check applicants. I believe the information is only stored for 48 hours, then automatically deleted. And, of course, the dealer does not submit information on the type, caliber or serial number of the weapon being purchased. Only your ID info is submitted for the background check.
So no, there is no national, or even statewide, registration.
Now, some localities may keep lists of their own. Although it's officially denied, I've had several dealers tell me that the Sheriff's Office where I live has a deputy whose fulltime job is going from one gun dealer to another, inspecting the 4473 forms and copying down the buyer's ID and gun info.