I have been to London several times and it is a huge city with tons of things to see and do. I would stay in London the whole week and as others suggested do some day trips. Have you bought the London Rick Steves guide book? Also look at the RS free videos and other videos for free on YouTube. Look at the RS scrapbooks on this website on the tours that go to London. Filled with lots of photos and ideas as to what to see and do. A week actually scratches the surface of all that London has to offer.
If you move to other cities, take into consideration, taking the time to get to a new city and checking into a hotel and learning your way around. I think you will be short changing yourself as to the time limit of just a week you have. Hopefully this will not be your first and only trip to London and the UK. You will go back.
You can go to Windsor Castle, easy train ride for the day. Go on the London walks, they have day time walks, pub walks in the evening, and day trips. Look them up on their website. London Walks and you don't need a reservation, just meet them at the assigned tube station and the walks are 10 pounds, days trips more money.
You can also take a boat ride up the Thames River and go to Greenwich for the day. Blenheim Palace and Warwick Castle are too far to do and require a tour to take. You could go to Hampton Court Palace by train from London. I did that and it was interesting.
Again, plan your days out and see what you can and know your going back.
Oxford and Cambridge you can take a train there for a day. But remember you only have 7 days and you want to see London. One or the most two day trips would be fine if you want to really see London.
London has plenty of sites to see and museums and pubs and plays.
Read up on what you want to see and do. And as you said you want to see cathedrals, there is St Paul's Cathedral and Westminster Abbey to visit. And for castles, there is the Tower of London and you can walk over the Tower Bridge and of course there are art museums. The National Gallery of Art, the National Portrait Gallery, the British Museum, the Victoria and Albert Museum, the parks to walk around in, the Wallace collection, the Tate Britian and the Tate Modern. And of course the food hall at Harrods.
This is just a few things I named. I am going to London for two weeks in May and am seeing all the things I have not seen and what I just named above, I have already seen but have lots more to see!
Have a great trip.