I’ve changed it to the new situation as from next Monday.
The north-western loop (Diane) is a two-way road open to traffic.
The southmost road (Boitsfort/Bosvoorde) is only for buses (shown as a dashed line in osm-carto because of the access restriction).
The rest is tagged as pedestrian roads.  Cyclists may use every road.

Current taggings ignores—and disallows—horses and emergency vehicles.  Still waiting for more accurate information about whether they can use the roads: pedestrian roads are currently barred with large concrete blocks, making it impossible for a fire truck to pass.

@Thierry1030 or others: you can have a look to see if it looks correct.