5m is pretty tall. We have grown some very tall Portuguese laurel, bought them at about 4 foot and they are now about 15 foot about 4.5 years later. Very easy to grow and prune, looks nice but nothing special.
I also bought some red Robins also 4 foot high when planted, pruned them aggressively at first and have made the lower branches quite bare so I can underplant, but let the tops grow up, they look great and must be 16 foot easily by now.
Because really I don't want laurels and red Robin's long term, i also planted in between - holly, Rowan, hawthorn, apple tree,, a flowering cherry, The hawthorn has shot up but is deciduous, ditto the cherry tree. Still, I dont use my garden so much in winter so the overlooking problem is less bothersome. The cherry tree is my absolute favourite - it is so beautiful in all seasons. I would definitely recommend.
The Rowan and hollies are going to take ages but eventually will be fantastic, I think. Again the Rowan is deciduous but will look nice and will disrupt views into my garden or house.
I would not personally plant leylandii as the maintenance is too much once they reach target height. Slower growing conifers might work, but then you will have to wait for them to get as big as you need, unless you have deep pockets and can afford a larger sapling.
If you are happy to prune every year, you could give eucalyptus a go! Grows extremely fast, and looks really nice.