On the GitHub side, everything will behave as if your new username had always been your name. All of your repositories will now belong to that new name and the old username will essentially no longer exist. This may take a few minutes to complete after your initiate the change.
This means that all references to the previous username, such as https://github.com/previoususername, will now give a 404 error. We cannot set up a redirect from the old username to the new one.
You will need to update the remotes in your local repositories to point to the new location. You can use git remote -v to see all the old remotes and look for references to your old username, then use git remote set-url <name> <new-url> to update them.
Your Git history is not affected and we should properly link commits to your new username so long as you don't also change your email address.