Posted: 26/08/2016 at 08:50
if the border is very narrow and clay it's going to be difficult to get anything which will provide the height you need. You could maybe install trellis panels in front of the fence and then grow climbers on them but this could be an expensive optionas you would need to provide supports so basically creating a second fence.
alternatively could you widen the border and grow a hedge on your side, I know it would take some time to provide the height but would provide a long term, easy care solution. If you do I would go for something you can clip back and contain easier than laurels, maybe something like beech.
My only other thought is that surely your neighbours don't want to look at bare leylandii either? Although it seems odd to have out the fence behind them. Maybe when they realise they won't regrow ( I'm assuming they're dead if they were hidden behind the laurels?) they will remove them.
Good luck in finding a solution