BitBucket Cloud: Enable Auto Fork Sync

Although I originally found this article, it looks like it only applies to their server product: https://confluence.atlassian.com/bitbucketserver/keeping-forks-synchronized-776639961.html

This article states that for this you will need to manually manage the local one: https://confluence.atlassian.com/bitbucket/forking-a-repository-221449527.html

After you create the repository, the original repository will likely continue to evolve as other users make changes to it. These changes are not reflected in your fork. However, you can bring these changes to your fork later by syncing the changes locally from the command line.

While this describes manually pulling upstream, perhaps you could script to do this more automatically for your own purposes. If I end up doing something similar for our team, I'll update this answer with more details, or maybe someone else will do the same.