I have deleted the name and added tags from the wood multipolgon.
The Mulgoa Nature Reserve seems ok. 
I didn't add the bicycle, motor_vehicle, motorcycle =no tags to the reserve as the access ways within as they seem to be tagged as needed, though I didn't check every one.
Normal practice is to add the access restrictions to the ways within protected areas instead of adding to the boundary multipolygon, but feel free to add them if you prefer.