Hi, welcome to GoY, the quickest growing screening plant that I can think of is ; x Cupressocyparis leylandii, yes the dreaded leylandii, I wouldn't recommend planting it, but it does orm a dense tall screen very quickly, the only trouble is that it can very qiuckly get out of hand, it can grow 3-4 ft a year, and as it gets taller, it gets more difficult to get to to cut it back, Derek.

Eugh! That's pretty ghastly! First thing would be to join forces with your neighbours and ask the council what they're going to do to screen your property.
Presumably you are past the stage of objecting to the plans - not objecting to the school in principle but objecting to the proximity to your property and the absolutely awful design.

How on earth did the council see this as a good idea!? so close to your house, completely over looking your garden, i agree with the leylandii, it is known as a "don't go there" type but unless the council are going to sort anything out then i don't see many other choices.

Thuja is nicer, smells better too

trellis and climbers is another idea, or how about a summerhouse to block it completly?
you could incorporate a greenhouse/conservatory into it

I would badger the council to provide a screen. The screen would then take up a bit of their land instead of yours and it would be their responsibility to look after it. The situation is so awful that this should have been a condition of permission to build.
Some friends of mine had a similar, although not as appalling, problem with an existing small country store obtained permission to build a very high extension behind their house. In the end the company agreed to plant a screen and my friends planted a eucalyptus tree which grew very quickly and has been judiciously pruned to encourage spread. The grey-green evergreen foliage is attractive and creates a good screen. They originally planned to take their tree down when the company's screen grew tall enough but now they love the tree!
I really feel for you! At one time buying next to a school playing field was a pretty safe option !

Is the Education Authority getting Govt. grant money for
this building ?
Best ask your M.P. about the financial aspect.
They have at the back of my flat, for the new Buccleuch
Academy building, only fortunately not so close.
The local Football Club obtained new changing rooms
in the old school building, new pitches, so everyone is happy.
The contractors obtained funding for a screen of trees, which have now been planted. They sent me a list of the names of the trees, and shrubs, it was all worked out by experts.
Just a question of asking the right people what is going on.

I'm just looking back through comments and wondered if you got any sort of response from the council on how they plan to hide their carbuncle.