Saved game files not transferring between Xbox one's consoles

I would think this has to do with an Xbox Live connection. The Backwards Compatible game is emulating the Xbox 360 and that includes the Xbox 360 Cloud Save system, which is a different pipeline from the Xbox One, though a similar idea. So when you're
in the game at your friends house, you should be able to press the "Menu" and "View" buttons (start and select) to pull up the 360 dashboard in the game, and then do some cursory functions via that dashboard, and possibly force the sync of the 360 cloud saves?
You might need to look at what storage device the saves are set to while on that console., but I can't remember if that is possible in BC games, as I think it only uses Cloud Saves.

So all that to say I think that it should be saving your 360 BC game saves to the cloud, but if it's not properly connected to Xbox Live, those saves might not be getting uploaded properly. I also remember those save being pretty big, so while you're in
the game, might want to check available space of your Xbox 360 Cloud Storage, again, different from your Xbox One or console storage.

EDIT: I want to point out that, while I was responding, ArminatorX described what I was trying to say in a much more clear and consistent manner

Just be aware that saving Xbox 360 backwards compatible titles is a bit more complicated than "native" Xbox One games.

The Xbox 360 emulated game has no concept of multitasking or being "suspended", so if you end your Oblivion session like a normal Xbox One game (e.g. simply turning off the console, or pressing the Xbox Button to switch to another game on the Start screen),
then the console might not be able to get the emulated game in such a state, that it can upload a consistent savegame.

If you want to make sure your Xbox 360 games are saved and uploaded properly, make sure that you save the game with the in-game save option, make sure that the save indicator for the game has disappeared, and then quit out of the Xbox 360 game by pressing
the Xbox button, pressing the Menu button on the main game tile showing the Xbox 360 game, and then choosing "Quit" from the menu.

When you quit the game like this, the Xbox knows the game is now in a consistent state to be synced to the cloud, and if your friends console is online, it should then sync your savegame as well to the cloud, so you should be able to continue where you left
off at your friends house.

Just make sure, that if you played on your own console in the mean time, and updated the save, that you don't downlaod that state to your friends console.

Instead play on your friends console and create a newer savegame, that should be "newer" than the last time your console synced.

Hi there,

just to be sure, your friends console is connected to Xbox Live?

If so then as far as I know it should synchronize with Xbox Live, but keep in mind that this can take some time.

If you still have a Xbox 360, can you check if you see the latest save game there?

Let us know the results

I'm not sure saves can be manually moved around to any extent. I know Xbox One games save to a cloud which syncs on any console you play the game on, but I don't think backwards compatible games do. Pair that with the complete inability to move saves between
storage devices, and, from what I can tell, you're pretty much out of luck. All saves save to Internal by default, regardless of any external drives or the default download location, and I'm not sure that can be overwritten. My best guess, and I'm
not even sure it'd actually work, would be to transfer the game itself from your drive to your friend's console, then back to your drive and hope that it brings your progress with it when it comes back. just know that I never have tried it, I've never needed
to transfer backwards compatible saves before. I'm sorry I can't be more help than to say "you can't", I hope someone else comes along with better advice. I'd be very interested to know if it's possible myself. Good luck, vod.

Xbox files are not meant to be read on a PC and transferred between a PC and console. If they're the files for the actual games, you would just have to redownload them. If they're saves, then they're lost unless you still have them saved on the console or backed up somewhere else.