A car will be extremely useful in the UK and Dolomites. Can't speak for the other areas, but I assume things are similar.
To further turtlespit's advice - Snowdonia *is* a rain magnet. The weather is often much better by the sea in Tremadog and there's heaps of awesome climbing there. However, at that time of year I strongly recommend heading down to the sea cliffs at Pembroke if you have time. It's a fantastic place to climb and is free of bird bans and army training in August.