I know your base will be London but if you can, spend at leat one day away from there. It's too easy to fall into the trap of thinking that you've seen the London so that's what the rest of the UK is like. It'd be like us visiting Walt Disney World and basing our whole idea of America on this. Try and get a trip to Brighton, Bath, Winchester, Norwich, York or, if you'r after a Royal kick, Windsor. The atmosphere in our smaller towns and cities is totally different to London.
Although I hated living there (another story entirely), York would be a great place to visit. From London it's only two hours on the train and you'd come out of the station to face the Medievil City Wall. Within a 10 minute walk of the station there is
York Minster,
The Jorvik Centre,
Cliffords Tower and my own personal favourite,
The Railway Museum. There are loads of other things to see like the Shambles (shopping & living as it used to be 400 years ago!), some great restaurants and you could top it off with cruise down the Ouse. York is a compact city and all the things I've mentioned are very close to the city centre and the station so a day trip would be feasible.
If you can get away from London during your stay in the UK then do it - we won't bite, honest!