Your second plan is much more doable, but always take into account that distances are probably greater than you imagine. Buses, Tubes and trains are good, but even so, they take time.

One tip: Try to book ahead for the train trip out of London. Go to www.thetrainline.com and get them to post the tickets to you at your hotel (having told them.) Booking ahead means you can get cheaper prices. However, I am not sure whether you will save that much. And, if you miss the train on your cheap ticket, the restrictions mean you might have to pay again. Check it out and see how much you might save.

The daily walking tour in the evening is a good idea. That way yuo see a lot and make the most of your tourist time.