30cm is only about a foot, which rules out free standing, taller shrubs, because they need to be 12, but preferably 18-24 inches away from any fence behind. Even Berberis 'Helmond Pillar', which is probably the narrowest one I can think of (although not evergreen) will not be as narrow as that, and will need to be over a foot away from a fence behind. Which means your only option is climbers, if you cannot widen the border. Evergreen climbers comprise a rather short list - Clematis armandii, Clematis cirrhosa, ivy, Jasmine nudiflorum and Trachelospermum jasminoides, which is certainly worth considering, although best planted in Spring rather than Autumn. You'll need a trellis or support affixed to the fence for it to climb up.
If you don't mind a much shorter shrub that you clip all the time, you might get away with dwarf box, keeping it clipped to form a small hedge to keep it from getting too wide, but again, that shouldn't really be planted too close to a fence.

You could use one of the columnar conifers such as Juniperus 'sky rocket' or the Italian Cypress (Cupressus sempervirens)

It would seem to make much more sense, and offer more scope, to lose a bit of lawn and give a decent depth to the border. How big is the lawn? Could you not sacrifice a foot of grass, at least in some places by re-shaping? You could add a lot of interest by bringing the border out into the lawn.

I have a very small fastigiate holly,(means it holds its branches almost vertical instead of spreading out sideways) very upright, very compact and narrow and slow growing, with tiny leaves. Mine is in a pot but it would be OK in the ground. Only trouble with using it for edging would be the expense.