tl;dr
cd "$ZSH" && git stash && upgrade_oh_my_zsh && git stash pop
Step 1: Navigate to the ZSH directory
cd ~/.oh-my-zsh
Step 2: Stash your local/untracked changes
git stash
Step 3: Run the automatic ZSH update
upgrade_oh_my_zsh
Step 4: Recover your changes
git stash pop