So, I have definitely never gardened in Britain, (though I have visited once
), but I hate to see a question go unanswered, so I will give it a shot and maybe someone else will see it and contribute more.
If you wall is truly full shade - like, along the south side of the yard, never gets sun all day long, you are going to have a tough time of it. I know taxus will grow in full shade, athough it will be a bit scraggly. It is at least evergreen, and you can get them in whatever height you like. I can't think of any vine that grows in complete shade other than english ivy, and I wouldn't really recommend that. Maybe some tall tropical annuals? (but here I am completely out of my depth, so hopefully someone else can advise on that front)
If it gets a few hours of sun, you can probably get by with trees and shrubs that grow in the under story, though, again, they may be a bit scraggly, and probably won't bloom as much as expected. If you're looking for wildlife benefit, something with flowers and berries will have the most impact.
I did a little research and it seems that Britain's native hazel (corylus avellana) is an important wildlife plant. The full-sized tree would probably be too large for your space, but it has a contorted version that it a popular garden plant which might manage?
Some other potential species I found. They have widely varying soil preferences which you would want to look into, but all seem to tolerate partial shade. I found them on this site:
Wayfaring Tree (Viburnum lantana). Different species of viburnum are frequently recomended in north america as having a good balance between wildlife benefit and attractive appearance. There are several cultivars. Similar case with Viburnum opulus (I just planted some of the american subspeices of that one in my wooded area!)
Cornus Sanquinea (redtwig dogwood) is another british understory plant with wildlife value and gardenworthy qualities. It's most famous for red branches which look good in the winter.
Rowan is also an under story plant with ornamental and wildlife value. Potentially too tall though.
Alder Buckthorn (Frangula alnus) is an understory shrub with berries
and another Buckthorn (Rhamnus carthatica) which is apparently an invasive weed in the americas
so maybe that means it would be sturdy and beneficial where it belongs? Don't know how attractive those last two are though.
Edit: It's funny how plant types are circumpolar. Virburnums and Dogwoods are some of the most frequently recommended wildlife plants here as well. It was fun looking up thing from a different perspective, and hope you can find something that works for you.
Oh, and another thought. Once a tree gets taller than the fence, it won't be shaded anymore, which should make it happy.