Fallout – Recover lost vault

The game has 2 different running in background behaviours:

• The game's wiki specifies that the resources will be consumed only for a little while after the game is no longer in use (game sent to background, phone not used etc). This is due to the fact that the resources require a lot of management and you would have to check/play the game every few minutes, half an hour at most, before your dwellers would start dying, get irradiated, etc
• Other activities that require a long time will still be in play (or generated once you open the game again, based on how long actually took since you last stopped playing - in normal hours) ... these activities are: wasteland exploration and returning, training, pregnancy, baby growing up, radio and creating resources
• You will also never be attacked by raiders, deathclaw or molerats while the game is on background.

As an example: A baby requires 3 hours in order to grow up and become an adult. When you open the game again, the game will check if 3 hours have passed since the baby was born to determine whether to make that baby an adult.

Warning: Players tried to fool the game by changing the timer, clock... be aware that this can cause lots of bugs, malfunctions, freezes, crashes etc

TL/DR: Yes, resources will be consumed only when you play the game or for a short while after you stop playing the game.

Your vault is not synchronised in the cloud.

Edit : since patch 1.2, it can be synchronized.

You have to enable Cloud save for your Vault in the load screen like so:

Manual backup

If you want to reinstall the game without using the cloud save, you can first make a copy of your save file - it is located in /Android/data/com.bethsoft.falloutshelter/files/VaultX.sav, either on your SD card or in the internal memory.