We enable git push on all instances by default. All you need to do is perform the initializing push, and then you can push changes easily. This is assuming you're working on a terminal.
Note: We only allow you to push files within the wp-content directory. Make sure your repo only contains the contents of wp-content, and not anything else.
Browse to your local git repository on the filesystem, and then run the following command:
git remote add timpani ssh://firstname.lastname@example.org/~/
This command is telling git that we're adding a remote server called 'timpani', and where to find it. We lock down remote access quite a lot, so the ending path "/~/" is vital. You can change "timpani" to be what ever you'd like to call our servers, and "instance-name" should be the unique name of your instance (the bit before your default .mytimpani.co.uk domain name. If you want to push to your 'uat' copy of WordPress, add "-uat" on the end of your instance name.
git push timpani +master:refs/heads/master
This pushes your entire local repository master branch up to us. Note: it will also automatically deploy what you push to your /wp-content/ directory.
Now, when you want to push further changes to us, just run:
git push timpani
Where 'timpani' is the name of the remote we setup above.