I'll go through all non-mammals now I have a bit of time, gonna be difficult to choose through all of this but will try and narrow it down 
Northern cassowary
[any kiwi species]
[Every penguin species we don't have
]
Blue-billed curassow
Blyth's tragopan
[any woodpecker species]
Spix's macaw
California condor
Kori bustard
Whooping crane
Siberian crane
Black-faced spoonbill
Japanese crested ibis
Kagu
Shoebill
[any bird of paradise species]
Wallcreeper
Yellow-billed oxpecker
Gharial
Chinese or Japanese giant salamander
Hellbender
Not going to state too many reptiles, as a lot will most likely be in the UK privately [not that mammals and birds won't, but some of them are much less likely].
Northern cassowary
[any kiwi species]
[Every penguin species we don't have
Blue-billed curassow
Blyth's tragopan
[any woodpecker species]
Spix's macaw
California condor
Kori bustard
Whooping crane
Siberian crane
Black-faced spoonbill
Japanese crested ibis
Kagu
Shoebill
[any bird of paradise species]
Wallcreeper
Yellow-billed oxpecker
Gharial
Chinese or Japanese giant salamander
Hellbender
Not going to state too many reptiles, as a lot will most likely be in the UK privately [not that mammals and birds won't, but some of them are much less likely].