I can recommend you to stay in city hotel. It is located close to the Tower of London and it is very clean and cozy. If you need more information about it you can check their website http://www.hitrooms.com/en/city_Hotel_gb3.htm. There you can check also for the prices.

Most of my times in the Cotswolds have been spent walking so distances may seem farther to me than if I'd been driving. But still, I wouldn't say that Winchcombe is "right next to Stow", 10 miles or more. But I will say that Winchcombe is a very nice town, have stayed there several times and Sudeley Castle is easy walking distance from the town. Winchcombe is also easier to get to than many places in the area - train to Cheltenham, local bus to Winchcombe. You can also walk easily from there along the Cotswold Way to Hailes Abbey, a lovely medieval ruin and Hailes Church across the road with it's beautiful and very old wall paintings.

I was amused at Janisj's description of Clapton as being "unreachable w/o a car" since I'm in my 60's and arrived there on foot. But she's right in the sense that it is a small, out of the way place which is, of course, part of it's charm. I believe it really is worthwhile, failing walking which is impractical for 2 days, to have a car to visit the Cotswolds. And you don't need to drive from London as it's easy to get closer by train and pick up a car locally.