Are these ‘official’ cycle routes? Are they signposted or marked on the ground in anyway?
Or is the only way of following them to use the tourist board’s map?

One of the main ‘rules’ of OSM is “map what’s on the ground”, so unless they are actually signposted I don’t think they should be added to OSM. As there is no way of verifying where the route actually goes.

Also, there may be copyright issues if copying these routes from another map. Do you have appropriate permission from the tourist board?

Even if they are not in the main OSM DB, its still possible to render them in Kosmos. eg you can add them as a GPX file.