I also  frequently add bicycle=yes (and foot=yes where there aren't separately mapped sidewalks) on London's "trunk" roads. I'd be happier if some routers actually paid more attention to the other tags, particularly low maxspeed=* values and the absence of expressway=*/motorroad=*, but until then...

On public bridleways, please do add tags!  Generally they should have designation=public_bridleway + horse=designated + bicycle=designated + foot=designated. You're probably already aware of them, but Robert Whittaker's PRoW resources can be very helpful.
https://osm.mathmos.net/prow/