I've recently lost my Vault save in unknown circumstances and that brought me here to search for solution. I did not receive any, but somebody posted a game dir for Android OS. That really helped. So after half an hour I've come with this solution:

1. Get into FileCommander (or whatever app that provides access to whole system folders)

2. Go to: Android/data/com.gamesas.falloutshelter/files (it can be both on SD card or phone memory, but in the built-in memory we can find a file we need)

3. Search for file named "VaultX.sav.bkp" (the X letter is a number of ur vault)

4. Copy it to some safe folder ex. "Downloads"

5. Plug ur phone to PC and open the same folder as u copied the .bkp file

6. Right click on the file, select Properties and change its name from "VaultX.sav.bkp" to "VaultX.sav" (I recommend u to do a copy of .bpk file before rename)

7 .Enter FileCommander and look for ur new "VaultX.sav" file

8. Copy and paste it into Android/data/com.gamesas.falloutshelter/files and replace the one which is already there

9. Enjoy ur Vault being back

P.S.

It worked fine for me on Android 4.4. Didn't test on any other OS, but the method would be similar I guess.

P.S. 2

Sorry for bad english