I gave this one to ZSEA ultimately based on the wider range of birds and mammals that they have that are native to mountainous regions and a slightly bigger conservation output. But it was still quite a close one in some ways and this is my take on it :
Range of species:
ZSEA
In terms of mammals ZSEA has quite a few native to mountainous regions or habitats across Africa, South America and Eurasia : Gelada baboon, Pallas cat, red panda, snow leopard, barbary sheep, king colobus (included if montane forests count under the category of mountainous), rock hyrax, Chilean pudu, Amur tiger.
In terms of birds ZSEA has a few from polar and mountainous regions : European eagle owl, snowy owl, Temminicks tagopan, Egyptian vulture, griffon vulture, great gray owl.
Reptiles / Inverts / amphibians : 0
Twycross zoo
In terms of mammals Twycross zoo has : vicuña, amur leopard, snow leopard , yellow throated marten, black and white colobus monkeys (included if montane forests count under the category of mountainous), Francois langur (included if montane forests count under the category of mountainous).
In terms of birds Twycross has three species that inhabit mountainous regions of North America , Asia and South America : Great gray owl, red fronted macaws, plumheaded parakeets.
Reptiles / Inverts / amphibians : 0
Conservation output :
Twycross : Doesn't seem to be heavily involved in conservation of species from mountainous regions but there is one project that it supports that is worth mentioning. Twycross is involved in funding and support for the Great Ape Survival Partnership which works with mountain gorillas which must surely count under the category "mountains". They have also been a long-term supporter of the mountain marmoset conservation programe in Brazil which gets them a thumbs up from me.
ZSEA : Is involved in supporting conservation NGO's working with snowleopards in Asia and the Wild cats conservation alliance which does work with big cats that include mountainous terrain as habitat like the tiger in Nepal / Bhutan / Himalayas and Amur leopards and tigers in Siberia. It is also part of the the Conservation Planning Specialist Group (CPSG) which does work with some species native to mountainous region.