We do have automatics now! And these days they are very good. Wouldn't for a moment suggest hiring a car in London though - public transport there is excellent (but not cheap)... but you may miss out on a lot of the Cotswolds without one - so many beautiful spots (previously mentioned, Bourton on the Water, Stow, Slaughters, Broadway, Chipping Camden) that aren't really reachable by public transport (I think there's a station in Moreton in the Marsh which is close-ish to Stow and Bourton on the Water). Broadway Tower (a folly) has a great view. Sudeley Castle (one of Henry VIIIs wives' I think).
Roads there are WAY easier than London - good A and B roads.
But you'll have a great time I am sure. Cotswolds are a lovely area. When are you going?