yeah, osm does not have schedule data, but it wouldn't be feasible to keep that info in osm, because it changes constantly.
instead bus schedules are supposed to be in GTFS datasets, normally published by the transport company. there are already services like transitous and motis, that aggregate and sanitize these sources for consumption by apps. comaps has plens for adding support, but it's going slowly.
https://github.com/motis-project/motis
Europe has relatively good coverage with these services, but as I see it the end user apps are still not ready. also, transport providers often don't publish realtime data, even if they are collecting it for themselves.