In case the Multisite functionality of WordPress is an option for you, then that would be the perfect use case for the template functionality of WordPress. You install your template on the multisite, “activate it for all sites” to be used and voilà, it should change on all “networksites”.
See the official documentation as a first start.
Besides that, there are hundrets of tutorials on the web and here on WPSE on this topic.
In case your sites are scattered all over the web and on different servers, then using .GIT repositories for this would be the right approach in my opinion.