Selecting the right Climber!

You've got two very different situations there, one south facing, lots of sun, and one west, much less sun, so you can either use two different climbers, or the only one I can think of that will be happy with either of those spots is Parthenocissus, virginia creeper. This doesn't cling initially, but once it gets going, is self clinging, but it's only appropriate if the bricks and mortar on your garage are in very good condition, not flaking or spalling. Ivy would work too, but does have more of a tendency to damage walls. Parthenocissus henryana is probably the best one to choose - this at least only gets 33 feet high. As you can see, you will need to regularly stop the growth at the top of the garage, or it will penetrate under any guttering and appear inside the building. If you choose to use it, plant it on the west facing side - any plant worth its salt will always head for the light, so hopefully it will cover both walls in time.
I have to say, were the garage mine, I'd not waste a south facing wall like that, though - I'd put up some support and grow clematis or a climbing rose or similar, with Schizophragma integrifolia, or Garrya elliptica, planted against the west facing side.

I agree with Bamboo put up some support (vine eyes and wire) and grow something a little more interesting…

I agree - provide some support and it will widen your choice.
I grow Eccromocarpus scaber on a west facing fence and it thrives. Not the tidiest growth habit, but hardier than it's given credit for.....

http://www.growsonyou.com/garden/show_garden_item/28088-chilean-glory-flower

Not sure where in the UK that you are, but how about Tropaeolum speciosum. Akebia is pretty hardy too.

I'd try White Jasmine on the south wall if your winters are not too severe. The Grandiflorum has a lovely subtle fragrance.