We're using Github Updater to manage this theme. View the readme docs there for more information.
Whenever you want you want to update this theme, make your commits and update the version number in the style.css header. Additionally, you can bump up the version tag to Git as well after you're done with your code changes.
git tag -a v1.0.0 -m 'My awesome tag'
git push --tags
The timeout is set to approx. 10 minutes, after which you should be able to directly update the theme through Wordpress.