I had the same issue. Can temporarily work around it by going to https://account.live.com/consent/manage and removing app permission from The Outer Worlds. Then, when it tries to login, click "no" when it asks for permission, and it will revert to the local saves. Or at least this is working for me.
Obsidian/Microsoft:
You need to fix this issue immediately. I would like to be able to play my game without having to spend an hour screwing around with Microsoft's incredibly obtuse "support" system before eventually stumbling into a temporary workaround on my own.
