Why not hire a bike?
Archimedes put up a post about bike hire here in the UK for you guys.
Otherwise hire a car, public transport (outside London) will not be what you find acceptable.
Is it one week in London then another on a culture cruise?. if so, my choice would be the south coast, Cornwall to Hampshire.
Still quite rural in places with picture postcard villages, remember the Romans were here, so steeped in history..
There are connecting highways, dual carraigeways/motorways, but the whole area is covered with good quality A+B roads.
Accomodation down there is excellent, from period contry inns/pubs, which generaly serve good value for money wholsesome food, right up to some of the grandest country house hotels in the UK, one of the real pleasures of my job, by the way, visiting and staying in these very same places.
Also at that time of the year it will have the best climate, especially if you are on a bike.
The cost of living over here compared to the State's might shock you, and the price of fuel is a disgrace.
You can get a book or two from the Tourist Information Office, I'll find out which ones, about places to stay giving all accomdation types and prices.
Compared to the US the UK is a spit in the ocean, but there is a hell of a lot crammed into it, so don't make time an issue, in other words just chill, after the hell hole that is London :-D , only joking.
Take your time, there is no point in doing 2-300 mile days, you would miss too much.
The UK is such a diverse but compact country with lots of different cultures, and it is easy to get from one end to the other in a day.
Huge sprawling cities to wilderness, a little known fact, the NW coast of Scotland offers some of the best secluded beaches you'll find anywhere, but best to keep your kegs (clothes) on in April.
Keep us posted, and enjoy your trip.