Is this your first trip to England? My guess is that it is, given the way your question was phrased.
"Is it worth traveling all the way to England from the US if one could stay only one week?"
Yes, absolutely. It would be far better if you could spend two weeks. But I'll take a week in England anytime I can get it! If you only have one week, I would not try to get to the the Lake District. If you have two weeks, yes, I would spend several days there.
There is enough in London to keep you busy for one week. You said you would be based in London, with day trips. That would work, to Cambridge, the Cotswolds, and many other places such as Bath, Windsor, Stratford-Upon-Avon, and out to see Hampton Court Palace. Or you could split one week, spending several days in London, then check out of your hotel, train to the Cotswolds, check into a hotel there for two or three nights.
Check out the company that Pam suggested, the London Walks Company. They do walking tours around London with a guide, and also day trips out to Bath, the Cotswolds, Stratford-Upon-Avon, and Cambridge, with a guide. Browse their website here:
www.walks.com
Their in-London walks cost £10, or £8 for seniors 65 or older, or full-time students.
This is definitely the way to do your trip out to Cambridge. Out of town day trips are a little more than in-town walks, but still a bargain. Here's the information on their day trip to Cambridge:
http://www.walks.com/Standalone/Cambridge_Tour/default.aspx