I own The Lego Movie video game on my Xbox 360, and my friends own it on the One. Is it possible to store a save in the cloud, which can be accessed by both consoles? I know that for backwards compat games it works.
The 360 save will not be in the same format at the XB1 save. Additionally, saves are tied to the gamer profile that they were created with.
I'm making the assumption that you and your friend are not sharing a profile, so even if it was the same platform, it could not be shared.
Another limitation is that the two versions will be seen as two separate games by the xbox systems as well - For example, I have both the 360 and XB1 versions of Metal Gear Solid V and as you can see below, they are treated as two separate games, even though they are the same, just on different platforms. The save data I have cannot be transferred here either.