Can original Xbox game saves be stored on the cloud?

Unfortunately no, only saves to the 360's hard drive will work as that has the necessary partition to run and save game files for the original Xbox.

I'm not sure how this information is stored for this particular game - whether it is on your console or on the Xbox Live servers - but if it is stored on your console, you would need to have your original Xbox 360 console available in order to recover these
stats. Your saved games can only be accessed on Xbox One if they were stored to the cloud storage. You can transfer all previous save data from your Xbox 360 storage device to the cloud storage by following the instructions here:

support.xbox.com/.../cloud-save-games

Someone else may have better insights into this, but this could be the solution to your problem.

The original Xbox games saves cannot be transferred. You will have to start those games again on the Xbox One if you want to play them. As for the Xbox 360 game saves, they can be saved to the cloud using the xbox 360, and they automatically transfer over
when you play a 360 game on the One. All saves on the Xbox One are stored on the cloud, and on the Xbox One. You can find the saves for each game by highlighting your game on my games and apps, press the menu button on controller and select manage game, and
from there you can find your saves.

Were your original saves stored on the cloud on your original console?

Try this suggestion;

The only way to manage Xbox 360 cloud saves on an Xbox One is from within an Xbox 360 game.

To delete it from the cloud you need to do what I said in my original post. Start the game, press both View/Start and Menu buttons together which will bring up the Xbox 360 dashboard.

Then go to Manage storage where you can manage all your Xbox 360 saves that are stored in the cloud. Deleting
Saves on Xbox One