They appear to be open to visitors on OSM, but are in fact sealed off. Also the peak is labelled ‘natural’ but sources say it was waste from a nearby gasworks, capped with clay from when the British Library was built. Maybe this info does not need to be added, I don’t know.
The tag natural=peak is described in the wiki as “the highest point of a natural or renaturalised physical feature”. We also use natural=water for ponds and lakes which did not form naturally and natural=wood whether it is the remnant of ancient forest or planted yesterday.
The default OSM Carto map tiles unfortunately render highway=footway + foot=private or foot=no as a red dashed line, not the pale grey dashes it would use for access=private. This issue was raised a very long time ago, see Render more specific access tags i.e. bicycle, motor_vehicle etc · Issue #214 · gravitystorm/openstreetmap-carto · GitHub