From Chris Babcock on the forums:
In Blueprint there is a “Control Screensaver” node. Add this with “Allow Screen Saver” checkbox off.
From code, use FPlatformMisc::ControlScreensaver(FPlatformMisc::EScreenSaverAction:isable).
Although there’s a comment there stating that it kills MoviePlayback on Android, I haven’t experienced problems (aside from seeking always jumping to 0 seconds, but that was already happening before).