Longleat has a small private area that the Koalas are sometimes allowed in
Edinburgh has off-show housing which the Koalas can access at any time, so would beat Longleat on this count.
There are two large outdoor enclosure,one of which has a pleasant stream running through it
Edinburgh also has a pretty large outdoor exhibit - larger, I believe, than both Longleat exhibits combined - which is sheltered from the elements and gives the inhabitants plenty of space and climbing area. There are no images in the gallery, but Google Streetview provides the following image viewed from afar:
