Certainly some of the larger mammals on display came from Bristol Zoo while the Sumatran rhinoceros and Thylacine came from London Zoo. I'm not sure about the smaller mammals and birds, they have so many of them that a large number must have been collected from the wild.
The museum it's self began when Lady Emily Greville Smyth donated her husbands private natural history collection to the city of Bristol. Sir John Greville Smyth was an avid collector and hunter but his wife was less enamoured with taxidermy and so gave the collection to the city upon her husbands death death. The museum was bombed during the Second World War and many of the larger items of taxidermy were lost, how much of the original collection survived I don't know but some of these smaller items may have originated from Greville Smyth.
Lady Greville Smith also donated the land that forms Greville Smyth. Eventually the whole their estate (Ashton Court) was sold to the City of Bristol.
I don't know too much about him and would lie to find out more. I've seen photographs of his collection and he a lot of taxidermy. He seems to have spent a huge chunk of his life travelling the world and hunting.
