Beech is not evergreen but it does keep its brown leaves all winter. If you want something that will look as neat as your present one you could consider Lonicera nitida, either the dark green one or Baggeson's Gold.

Box isn't worth risking because of box blight which seems to be becoming more widespread.
Yew makes a good dense hedge but is very dark and can eventually easily get very wide.

Just had a look at Lonicera Nitida on line. It looks exactly what I'm looking for - dense growth like box, fast growing and evergreen! Thanks for that suggestion Steragram - think you've solved my problem - unless someone else has an even better suggestion!

I'd have gone with Lonicera nitida too - but there is a newer variety with variegated leaves called 'Lemon Beauty' which is very attractive, but is said only to reach a metre in height, so probably not tall enough.

Thanks to both of you. Decided to take your advice and go for Lonicera Nitida - the green type.

Choosing the right hedge is more about which one do you like. You have many options. Take a stroll over to your local garden shop. They'll have a selection of many possibilities. I have no idea what you would like.

Privet is another option, but it's semi-evergreen (only keeps it's leaves in mild winters).

I wouldn't go for box due to the dreaded box blight.