When you look at a game's local saves on xbox, the default option that is selected is...

I understand the frustration and I'm sorry that this has happened to you. It may not be a bad idea to provide this as feedback to the concerned team by following what's mentioned here => Provide feedback as an Xbox Insider | Xbox Support.

Disconnect your Xbox from the network so it is offline.

Load Fallout

It should default to looking at your local save files. Load the latest one and you should hopefully see your save with latest progress.

Load up that save file and then reconnect to the Internet.

Then save you game again, you might see a version difference pop-up and make sure you select the option of using your local save as the primary and the cloud should get updated with that correct save.

You may need to load up a few of the save files to find the one you want as Fallout 4 creates multiple saves.

If the save isn't available locally, it is likely your progress is lost unfortunately.

Thank you. If the disc stops when you're in the process of installing it kindly clear the locally saved data then try again.

1. Eject the disc from your console.

2. Press the Xbox button to open the guide.

3. Select Profile & system > Settings > System > Storage devices.

4. Select Clear local saved games.

Note Selecting this option will not delete any games or progress you have saved in the Xbox cloud. It will only clear the data you have saved locally.

5. Select Yes to confirm.

6. Once your console restarts, try installing the game again.

When you go to Settings - System, on the right there will be an option for Storage. Select Storage and then click on "Clear Local Saved Games".

Hi Goldcrane! Sorry to hear about your issue. Let’s do our best to sort that out.

What exactly is your Xbox console model? When did this issue start to occur? Are you trying
to download a game digitally or through disc?

Let’s try to clear your local saves:
Press the Xbox button to open the guide.
Select Profile & system > Settings > System > Storage devices.
Select Clear local saved games.
Note: Selecting this option will not delete any games or progress you have saved in the Xbox cloud. It will only clear data you have saved locally.
Select Yes to confirm.
Once your console restarts, try installing the game again.