welcome to GoY.
my first thought was to add a trellis and then plant evergreen climbers. there are examples of honeysuckle and clematis that if selected carefully you can have some in flower most of the year. a climbing rose would also help screen it as the main stems are usually tied in horizontally.

how much soil do you have your side of the fence. this will be important for what ever you choose.

Well, that's one reason I love my Spartan Junipers. They grow about 15 feet with a narrow profile and clusters of blue berries that can be used in the kitchen or if you are feeling ambitious make gin. The junipers don't take up much real estate, makes a wind break, attracts song birds and always smells like Christmas. If you have well draining soil - like most trees & plants, they must have well draining soil. I would easily recommend these. No muss, no fuss evergreen provides a solid green wall of privacy.

If you go for conifers do check the regulations in your area - some UKcouncils have height limits because of possible shading of adjoining properties, which is why some people compromise with trellis!

I believe that even with trellis added to the top of a fence it is necessary to get planning permission if this puts the height to over 6 feet. Variegated Holly is also a possibility, there are some nice ones, silver & golden types.

Welcome to GOY from me too. I am also close to Cambridgeshire - borderline Norfolk - and on clay. Great stuff for nutrients and the roses love it, not so good for digging! We have just had huge Leylandii hedges cut down to about 8 ft each side, enough to stop the wind a bit, but leaving bare trunks of course. Many are already covered with native Ivy but I am planning to cover the rest with climbers, Clematis, Passion flower, Jasmine etc. I also have some Goldheart ivy which should brighten them a bit. Anything is an improvement on 30ft Leylandii full of pigeons!