RRonning84, aka Ryan the husband, said I need to log on and answer this since I've been to London a multitude of times...
The London Tube will be your friend! You can see Westminster Abby, Buckingham Palace, Kensington Palace, and the British Museum (go see the Egyptian collection!!!) all by taking the tube to the correct location. There are maps available from the Tube that show all the major sites and what subway line they are one. Don't miss the tower of London and tower bridge....the view from the top of tower bridge is great!
Although I'm sure you'll want to eat in local restaurants and pubs, buying food at ASDA (their version of Walmart) is far more economical than spending $20 in a pub for a salad or $18 for bacon and eggs.
If you have time...there are bus tours available to sites such as Stonehenge, Bath, and the "Harry Potter town"..... these don't really have to be booked more than the day or two before.
Four days just isn't enough...so plan your siteseeing in groups of places that are close together.....
And, don't forget to shop at Harrods! (all around that area are great shops and restaurants too!)
If you want to spring for a ticket for me....I'll go along and show you around!