Contrary to what most have said, I would encourage you to stay in London. You seem to have plans for only one day in London which is a very short time to spend seeing much in a city as large as London. Here is just one walk you might consider. Visit the Tower as your first stop on Sept 4th and after you visit the Tower, walk across the Tower Bridge. Visit Borough's Market (bring a camera), it is a food and people scene. Lots of people enjoy grabbing something there for lunch; you should too. Try Scrumpy! Stop at Southwark Cathedral, whose history dates back to before 1086. Then set off on foot down Bankside/Jubilee Walkway ...stop at the Anchor and have a half pint, right on the river with outside seating. (btw, all this walking is on a pedestrian only wide walkway) How about a quick visit to The Golden Hinde, a replica of the famous ship of Sir Francis Drake (c 1577) Walk by or stop in the Globe and also the Rose, two theaters from Shakespeare's time (the Globe is a recreation and does have tours and performances). Walk on to the Tate Modern.....not my fav but it is free and you are there so spend 1/2 hour looking around. Look at the Millenium bridge or walk over to St Paul's. Just past the Tate is the Bankside Gallery, which sells paintings, large and small, inexpensive and very expensive, lovely stuff and perhaps souvenir material. Stroll on. See the iconic OXO Building. and BlackFriars' Bridge. A little further look across the River to the gardens of the Inner and Middle Temples....the legal center on London at the Inns of Court. Haven't been.....try a tour from http://www.walks.com. More on them later. Admire the elaborate lamp posts (with cod). As you get closer to the EYE, all kinds of things are happening on the Walkway, seemingly anchored by the SouthBank Center. Look down the River on the other side and enjoying the views as you get closer to Westminster Bridge and the Houses of Parliament and, of course, Big Ben (we know, the bell is Ben, the tower is really the Queen Elizabeth Tower). Ride the Eye or cross the bridge and 1. stop at St Stephen's Tavern Pub, serving real ale and right on Bridge St 2. get on a boat running back up to the Tower so you can see it all from the water, 3.go on to Westminster Abbey, 4 take a pre-arranged tour of the Houses of Parliament,
For heaven's sake, take some London walks. They are very inexpensive, they have a full and amazing schedule, you don't have to reserve, just pick what you want and show up at the meeting spot AND they cover small areas giving you a view you would never have otherwise I love illegal London (the Inns of Court), Old Westminster and the City.
Stay in London and get to know it.