I am trying to get bicycle routes data (vector data/shapefile) of major cities of Europe.
Can someone tell me how to download the bicycle routes shown in the OpenCycleMap map?
OpenCycleMap data are available as a layer on OpenStreetMap. In fact,
The OpenCycleMap global cycling map is based on data from the OpenStreetMap project.
The map is updated every few days with the latest data from OpenStreetMap.
Thus, see this similar question on help.openstreetmap.org.
You can try Overpass API, if your area is small enough. Basically, the query should be like this:
[out:json][timeout:90]; area["name"="Københavns Kommune"]->.a; ( way["highway"="cycleway"](area.a); rel["route"="bicycle"](area.a); ); out geom;
Link to the query, select Settings > Map > Don't display small features as POIs, if you want.
You could also ask your question on gis.stackexchange.com.