I set a switch-on time and nothing happened

Settings → Switch on at only takes effect once the mode is enabled. Press Enable coming soon mode (or Enable maintenance mode, or Close the shop for orders) on the Dashboard: the Dashboard then says the page is scheduled, and visitors see it from that time. With the mode off, the Dashboard and the Settings screen both say your page is set to go up at that time once the mode is enabled, and the Dashboard has the Enable button right beside it. If the time has already passed, they say so: set a new time, or press Enable to put the page up now.