London is a shopper's mecca, with stores hundreds of years old and literally everything else. If the world's most determined shopper shopped all summer long, they could never hit all the stores that London has to offer. Focusing on a few famous streets and their environs will perhaps give future tourists an idea of where they'd like to start their shopping spree.
Ask any American of a certain age, and they know who they are, about Carnaby Street. This became a very important street indeed, as in the 1960s, American kids tried to look and dress as cool as their British cousins. Carnaby Street is pedestrian; It sports an arch welcoming all to its area, so it's hard to miss. It's north of Soho and can be reached from the Oxford Circus tube station; just behind Regent Street.
The trendiest part of Carnaby Street these days is Kingley Court. There will be found one of a kind concept shops and studios. There are fashion, music, and things you can't find anywhere else in London. Stop and have a pint at Shakespeare's Head, established in 1735. You'll find pub grub, as well as a restaurant.
Oxford Street is said to be Europe's busiest shopping street. It does hold many of the oldest and grandest department stores; Selfridges, Marks and Spenser, Debenhams, and House of Fraser to name a few. It is 1 1/2 miles long, with many tube stations interspersed. Many of its cross streets, such as Tottenham Court Road, have many shops and eateries well worth investigating.
Piccadilly and Jermyn Streets are on the wide and busy road that leads to Piccadilly Circus. Fortnam and Mason sells many exotic foods, and has for over 300 years. Buy some for a picnic. Check out Hatchard Booksellers, established 1797. Piccadilly Market is worth a detour. Burlington House, home of the Royal Academy, holds summer art shows; a break from shopping for a little culture won't hurt. Really.