Technically they will likely tell you to cancel and re-file. This of course sucks because it moves you to the back of the line.
Your approval or disapproval will be returned by mail to the address on the form 1, and the USPS will not forward correspondance from the ATF to your new address, so unless you are on good terms with the current resident there, it will be lost at that point.
After a time you can call to inquire the status of your form. If it is approved, you can request a certified copy of the form mailed to your new address, but it will not have a physical stamp, if you value that, but is still valid.
After all that, when you actually build your SBR, be sure the city/state engraved is where you actually assemble it, and while it's not required since you moved intrastate, the ATF would request you file a form 5320.20 for change of address to keep the registry up to date.