Not to be pedantic, but OSM is just data, you're asking whether there exists software that can do this. (You might for example ask in the OsmAnd Telegram chat whether this can be done with OsmAnd.) I don't know if there is any that sticks to only cycle paths, but there is definitely software that'll avoid car-only roads like highways, try to avoid big roads where reasonable, and prefers cycle paths where available, such as OsmAnd (you can also configure routing preferences like shortest way, prefer byways, configuring your cycling speed, using elevation data...). For turn by turn instructions, that seems to be pretty standard and OsmAnd can do this too. Have you tried any software that uses OSM data?