I was in London this past March on business for two weeks and really lucked out on the weather. My wife and I went to the British Museum a couple of times, the Tower of London was good and the Victoria and Albert Museum was also good.
We were standing outside of the Ye Olde Cheshire Cheese pup near Fleet Street and two different locals said we had to go in to experience that pub. It was really unusual and worth getting a beer to take in the locals.
http://https://en.m.wikipedia.org/wi...heshire_Cheese
We felt very safe in London too.
And one last tip. Use the subway (the underground or the Tube as they say). Just tap a credit card on the turnstile going in and once you get to where you are going tap the credit card again to leave. Very convenient and fast.