Is there such tag timetable?
Most timetables will not be licensed in a way that allows them to be used. You wouldn’t, for example, be able to do this in London.
I think the other issue might be that the information would be unmaintainable. Even in London not every bus route has been mapped yet, and I doubt that those that are mapped are regularly checked for still using the same route. Stop names can take years to get updated when they change. Tracking small changes in times may be even more difficult.
Many transit companies/agencies publish their schedules in GTFS format https://en.wikipedia.org/wiki/General_Transit_Feed_Specification. Some of those agencies have reasonable (open or no cost) licensing. And those that have open licensing can be aggregated at https://navitia.opendatasoft.com/explore/?sort=modified. From there it can be used by a number of routing apps like https://transportr.grobox.de.
Given how many different transport agencies there are in the world and how often they change their schedules, it seems to me that this is data best maintained by the originating agencies and not kept in OSM.
Maybe have alsoa look at https://www.graphhopper.com/
they have integration for GTFS data for public transport