Note: this list is generated by looking for the string for this theme hook, so it may include some references that are not actually using this theme hook.

theme_update_manager_update_form in core/modules/update/update.theme.inc
Returns HTML for the first page in the process of updating projects.
theme_update_report in core/modules/update/update.theme.inc
Returns HTML for the project status report.
update_cron in core/modules/update/update.module
Implements hook_cron().
update_theme in core/modules/update/update.module
Implements hook_theme().
update_uninstall in core/modules/update/update.install
Implements hook_uninstall().
_update_process_fetch_task in core/modules/update/update.fetch.inc
Processes a task to fetch available update data for a single project.