A merged contribution to openstreetmap-carto is not automatically published. The maintainers periodically tag the latest set of updates and publish it into a GitHub release.
To verify the current version of openstreetmap-carto, check the Releases section of the GitHub repository. Besides, the main changes are included in the CHANGELOG document. The latest release is published to the tile servers. The OSM server administrators can control which version to install1 by manually updating the related Chef role, where the selected version number of Openstreetmap Carto is reported.
The update process foresees the re-rendering of z0-12; then any tile older than the style file is treated as dirty so that it will be re-rendered (if possible) when it is next requested2.
There are the following three tile servers at the time of writing, with link to the related hw configuration:
Openstreetmap uses Munin for server monitoring,
Tile server munin graphs are available to tell what’s happening on the systems; there is also a description on how to interpret them.
Summary of the software configuration of the tile servers as found in some GitHub posts and sites:3 4 5
The Operations Working Group log reports updated information regarding versions.
Other information (from Munin):
Other useful references: