Download BIKE maps as vector/.shp/ something I can use in Illustrator?!!

OpenCycleMap is a personal project of Andy Allan (Gravitystorm), the styles used to create it are not available, and even if they were would not be much use for pulling into Illustrator.

You will need to use a basic OSM extract (such as those available at BBBike) and select the features which you require either using various OSM tools, such as Osmosis of osmfilter, or using ShapeFiles in ArcGIS or QGIS. Basically you will need at a minimum all highways tagged with highway=cycleway, or cycleway=yes or (highway=footway AND cycle=yes) ncn|rcn|lcn=* or ncn_ref|rcn-ref|lcn_ref=* or cycleway:lane=*. You may also wish to see barriers, cycle parking, cycle crossings etc. You are unlikely to replicate anything like OpenCycleMap because it represents at least 5 years of effort by Andy, but you may get a reasonable approximation. Note that tagging of cycleways varies according to local circumstances.

Printed cycle maps have been produced from OpenStreetMap data, for instance by David Earl ("What I learned making a real map on real paper for real people and real money", David Earl at Sotm2010, Girona). If you watch the video you will see that it is a complex and involved process, not something to do at the last minute.

I am sure that one of the OSM service providers would be able to provide you with a suitable extract, but I imagine not at a price you would be comfortable with.