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Although adding bicycle=no on the N-S highway=footway does no harm, it doesn't actually do anything either as it's implcit.
https://wiki.openstreetmap.org/wiki/OSM_tags_for_routing/Access_restrictions#United_Kingdom

For the E-W highway=cycleway, if cycling is not permitted, you should probably change the highway type. Looking at the Bing street side imagery it seems to be wide enough for service vehicles and has a removable bollard, so perhaps:
highway=pedestrian + vehicle=private 