I had to do this months ago because I got a new3DS and my old3DS died
it is possible to use that backup SD data on another new 3ds
(I mean a new one, not the "new 3ds" model)
but you'll need a copy/backup of the NAND from your stolen 3DS and the OTP.bin file
it's pretty simple, if you open the CTRNAND on a 3DS and go to the "private" folder, you'll see a file called "movable.sed"
this file is the one that is transferred when you do a System Transfer to another 3DS
-first of all, use GodMode9 and open CTRNAND SYSNAND, "private" folder and check the movable.sed file, check if it's 320 Byte or 288 Byte and take note of that, also,
backup this movable.sed file to the SD card
-on PC, use fuse3DS (download if you don't have it) to mount the stolen 3DS NAND image to a folder,
you need the nand.bin and otp.bin for it to work
-now open WinImage (download if you don't have it)
-open the ctrnand_full.img file using WinImage
-open private folder
copy movable.sed to the new 3DS SD card
+if this movable.sed file matches the size of your movable.sed file on your new3DS, continue with the tutorial
++if this movable.sed file does not match the size (i.e. your stolen 3DS have a 288 byte file and your new 3ds have a 320 byte file) you'll have to modify it
*if your stolen 3DS movable.sed is 288 byte,
fill with 00 the last 16 bytes so it's 320 byte now and it matches your new 3ds movable.sed size
*if your stolen 3DS movable.sed is 320 byte,
change the byte at offset 0x5 from 01 to 00 so it's 288 byte and it matches
-make a backup of the new 3DS NAND
-after you backup the NAND open GodMode9 and copy the movable.sed you extracted from the stolen 3DS NAND to SYSNAND CTRNAND>private and replace it
-after it's done, if you reboot, you'll see the 3DS setup screen because when you replace the movable.sed, the 3DS formats itself, but don't worry, setup your 3DS and go to the Home Menu
+if instead of the setup screen you got an error like this one,
make sure the extracted movable.sed is the same size as your new 3ds movable.sed size
-now you will be able to use almost every app you had on your stolen 3DS, if you don't see a game, you have to install a ticket for that game and it should work
-also, some games (2015+) probably won't open (stuck on 3DS screen) but it's easy to fix, go to FBI, titles and import seed for every game you have an error (or import seed on EVERY game like I did) and everything should work
I don't think I missed something
good luck, make sure you always make a NAND backup