Make the time-out configurable by the tests. In a config-file on the test environment, a setting in a database or passed as a parameter in the API calls. Figure it out, because test-suits should be fast and automated, so there should be value in adding a features to configure it. Make it testable by automation, or get a development team todo it for you.
I am the maintainer of a open-source Chrome extension called Stoppable. It auto blocks websites after a couple of minutes. All the end-to-end Selenium tests set the default value of 5 minutes to two seconds. The whole end-to-end test suite runs in 30 seconds, not minutes. Making it quick to verify all functionality still works and it gives me confidence to release it without any manual testing.