This corner of my garden is shot from indoors. It gets zero direct sun. This bush (a pyracantha?) clearly wants to be much larger than my predecessors let it, but it is now just blocking the view. If I cut it back we get no flowers. Meanwhile, the fence is ugly but a way down the priority list for replacement.
I am not keen on birginias but the oh loves them so they are going nowhere.
I am a big fan of light indoors, so I'd be tempted not to put anything large in front of a window. I'd ruthlessly chop it down and instead use some small, delicate, low- and medium-height shade loving things to give interest and colour all year. I'd cover the fence with flowering climbers to give a lovely green backdrop.
Is it wet or dry shade? There are plenty of lovely plants for both!
I'd say it's pretty dry, there's a privet the other side of the fence (next door keep it in decent order) so I'd guess that sucks up moisture. I've not really ventured into it much so far, my defective cat was using the corner as a toilet, and a huge pile of rubble from other excavations made it a bit inaccessible all last summer. There's a hideous bog standard dogwood the far side of the bush, that is coming out for sure.
I would love climbers, although my efforts last year with morning glories and spanish flags failed dismally. I'm a bit woried about toppling the fence, but it takes full grown foxes vaulting over it on their fox superhighway, so it can't be that flimsy!
The thing about those climbers is that they are annuals in a British climate. So they were always going to fail at the end of the year! Going for something that is perennial, even evergreen will be a very different story and will give you colour and interest for much longer!
If you put up wires (check out Gripple - easiest system ever to use) then you can ensure they grow horizontally on your side and really cover the panels. They might slip over to next door's too, but they can keep them in check or let them wander, as they see fit! You definitely need a bit of clearance from the grass around any kind of climber, especially when getting them established. Grass will out-compete most things!