I would pick the Lake District over Scotland especially for the level of walking you want to do. I love walking in the Yorkshire Dales. Most Americans probably go to the Cotswolds-easier terrain there. As for the weather, it is Great Britain so you can always expect some wet weather and if it is sunny the whole trip just be thankful and happy. Be sure you take proper equipment including full waterproofs in a breathable fabric and comfortable broken-in boots with good support and waterproofing.