When you say 'exposed' do you mean that there are strong winds where you are? Or simply that it doesn't get any shade? I'm assuming you're in the UK, so the sunshine is British sunshine, not the baking heat of Crete!
Loads and loads of climbing plants love a really sunny site. Roses would be one, abutilon another, Tropaeolum another. Actindia is a stunning choice for deciduous foliage. For evergreen, trachylospermum jasminoides, clematis armandii are a classic choice- a more unusual one, if you have the space, is a magnolia grandiflora grown as a climber.
However, if it were my house there's just one thing I'd choose for a site like that, and that is a wisteria. Now this is a slow burner - it's the kind of thing that takes years to get going. It also requires a bit of knowledge and care. And it won't give you evergreen leaf. But it is, quite simply, the most beautiful plant in terms of its windy, odd, twisting bare branches over winter - and the most stunning and fragrant when in flower.