In that case, you would probably need to fork Unreal Engine into your account under the Epic Games organization AND create a private repository in your own organization that you ALSO pull to your local file system.
You can add that first repository on your local file system as a remote for your second, private repository to use Git to manage merging and such. Someone would need to pull from Epic into your first repo, then pull from your first repo into your second repo (handling any conflicts of course), then push to your private GitHub repo.
For a semi-related example, see: http://treeleafmedia.be/blog/2011/03/creating-a-new-git-repository-on-a-local-file-system/
Obviously, your personal GitHub repo cannot be a public repo.
Epic might want to add to or correct this. I am obviously not affiliated with them.