Is this laurel on your land or your neighbour's?

I can't see why you can't plant whatever you like if it's on your own property and not a conservation area. You need evergreens for you privacy and Thuja Plicata would be fast-growing and make a neat hedge. As suggested on the other question by Bamboo, Taxus Baccata would be even better if you have time. If the council or whoever in involved then I would try Cotoneaster Conubia is a semi-evergreen.

Just had another thought - have a look at Lonicera nitida.
It isn't a native but they might not notice.Eminently clippable and will grow reasonably tall if you let it. Very compact with evergreen tiny leaves. Holly is native but fairly slow growing initially and horrible to clear up after pruning...