In reply to CurlyStevo:
Another vote for the Dolomites - or alternatively either the Kaisergebirge or the Wetterstein. I have climbed extensively in all three areas and any one of them would be ideal for your requirements. I have also climbed extensively in the Ecrins (La Berarde and Ailefroide), Bregaglia, Stubai, Central and Eastern Switzerland as well as the Karwendal but would recommend the first three over these latter areas.
No need for axes and crampons at any of them. The Kaisergebirge and the Wetterstein are somewhat 'below the radar' these days but for trad multi-pitch they take some beating. I suspect that you'd be lucky to meet any other Brits at either of them.
However, if I had to choose, it'd be the Dolomites every time. I've been there in July/ August and the honey pots - Vajolet and Sella Towers, Passo Pordoi, Cinque Torre for instance - will be quite busy. But it's very easy to find equally good climbing in much quieter spots; places where you can have a whole route, or even a whole mountain to yourself.
HTH
Dave