I haven't been to many different zoos (though hopefully that will soon change). But from memory, some bird species I have seen that I'd deem relatively rare/unusual include:
St Lucia Amazon - Pairi Daiza (the only place it's kept worldwide)
St Vincent Amazon - Pairi Daiza (rare for Europeans like me but I'm aware Houston and San Diego keep them as well)
From Pairi Daiza I would add Spix's Macaws and Long Crested Eagle but unfortunately I missed out on them on my visit. From Chester and London, nothing springs to mind for birds, and I don't really keep a good track of other animal groups properly from other zoos (something else I need to get better at).
By the end of this summer, though, this list (definitely for birds at least) should improve, as I'm visiting Newquay, Paradise Park, Hawk Conservancy Trust, and Chester (again):
From Newquay, I should see Chinese Grosbeak (found nowhere else in captivity) and some subspecies of Vasa Parrot (plus Yellow Cardinal, Cuban Grassquits, and Bearded Tits, although I don't know how common or rare these two are for captive institutions).
From Paradise Park, I should see Lear's Macaws (I think they're quite rare), Collared Falconet (kept nowhere else globally), Azure Tit (only found in a few other places), Rajah Black Lory, and a wealth of other species.
From the Hawk Conservancy Trust, they do have some pretty special ones, like Wahlberg's Eagle, etc.
I do want to travel to mainland Europe fairly soon to cross off other zoos from my list (mainly in Germany, Czechia, and zoos dotted about in other countries) as well as a return trip to Pairi Daiza in Belgium to see those which I missed last time.