Everyone seems to forget that if you prune a hedge that inhibits potential root formation. In heavy clay Beech is not a good idea so I would use Hornbeam. It keeps it leaves and looks smart when established (at least 3 years after planting depending on initial plant size).
Hornbeam can be kept less than 20cm thick, roots are not a problem (in clay soil you want it to soak up water much of the time), so that is the hedge plant that I would suggest. It does look very similar to Beech and also keeps its leaves.
For privacy laurel is good but it gets very thick, to thick unless you have 1-2 metre at least to let it get thick. Thuja is also worth a look at - it takes a good prune and is a great privacy screen, Bamboo is also another good screen for urban gardens and it is light enough to let light in.