highway=cycleway is for ways designated primary for cycling (meaning there is usually a cycling-specific sign and or paint markings, etc.). See https://wiki.openstreetmap.org/wiki/Tag:highway%3Dcycleway#Foot_access_on_cycleways

"Cycling is not forbidden" is a legal thing and should be tagged as "bicycle=yes"  (e.g. "highway=footway bicycle=yes"). However if its by default allowed to ride a bike on footways in the whole country then no need to tag "bicycle=yes" on every sidewalk. Its already known, see 
https://wiki.openstreetmap.org/wiki/OSM_tags_for_routing/Access_restrictions (and is common in many countries).

Its also possible to use "highway=path foot=designated bicycle=yes", see https://wiki.openstreetmap.org/wiki/Path_examples
