Nominatim does NOT search route relations. I use taginfo & overpass.

The downside of overpass is that it:

• does not use wildcards by default using the wizard
• does not search case-insensitive by default using the wizard

Example of searching for the route Härnö Trail in Härnön: https://taginfo.openstreetmap.org/search?q=h%C3%A4rn%C3%B6+trail#values -> click the orange -> click overpass symbol -> from overpass you can click the map object and -> click to osm.org from there

Alternatively you can download Härnön in JOSM and search for the relation there. (much easier if you know where it is beforehand)

iD does not yet show relations, only memberhip of relations on objects. Also iD lacks a way to check that your edit did not break (the continuity of) one or more relations.

(edited to remove the incorrect assumption that it does not accept wildcards and to specifically mention route relations)